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className":1240,"style":887},[886],[19,1242],{"className":1243,"style":865},[225],[19,1245,1247],{"className":1246,"style":894},[195],[19,1248,712],{"className":1249},[195],[19,1251,1252,1255],{"style":900},[19,1253],{"className":1254,"style":865},[225],[19,1256,1258],{"className":1257,"style":908},[907],[318,1259,1260],{"xmlns":320,"width":911,"height":912,"viewBox":913,"preserveAspectRatio":914},[327,1261],{"d":917},[19,1263,241],{"className":1264},[240],[19,1266,1268],{"className":1267},[213],[19,1269,1271],{"className":1270,"style":927},[217],[19,1272],{},[19,1274,241],{"className":1275},[240],[19,1277,1279],{"className":1278},[213],[19,1280,1282],{"className":1281,"style":1111},[217],[19,1283],{},[19,1285],{"className":1286},[674,675],[19,1288,1290],{"className":1289,"style":979},[674,978],[19,1291,741],{"className":1292},[281,983],[19,1294,1296],{"className":1295},[204],[19,1297,1299],{"className":1298},[208],[19,1300,1302],{"className":1301},[213],[19,1303,1305],{"className":1304,"style":1135},[217],[19,1306,1307,1310],{"style":1138},[19,1308],{"className":1309,"style":226},[225],[19,1311,1313],{"className":1312},[230,231,232,233],[19,1314,52],{"className":1315},[195,199,233],[19,1317,1319],{"className":1318,"style":979},[674,978],[19,1320,741],{"className":1321},[281,983],[10,1323,1324],{},"With my current knowledge, I can only comprehend two proof methods as follows:",[1326,1327,1329],"h2",{"id":1328},"matrix-method","Matrix Method",[10,1331,1332,1333,1386],{},"Let's first discuss the case when ",[19,1334,1336,1354],{"className":1335,"translate":23},[27],[19,1337,1339],{"className":1338},[31],[33,1340,1341],{"xmlns":35},[38,1342,1343,1351],{},[41,1344,1345,1347,1349],{},[47,1346,52],{},[54,1348,171],{},[81,1350,158],{},[175,1352,1353],{"encoding":177},"n \\geq 2",[19,1355,1357,1376],{"className":1356,"ariaHidden":61},[182],[19,1358,1360,1364,1367,1370,1373],{"className":1359},[186],[19,1361],{"className":1362,"style":1363},[190],"height:0.7719em;vertical-align:-0.136em;",[19,1365,52],{"className":1366},[195,199],[19,1368],{"className":1369,"style":255},[254],[19,1371,171],{"className":1372},[259],[19,1374],{"className":1375,"style":255},[254],[19,1377,1379,1383],{"className":1378},[186],[19,1380],{"className":1381,"style":1382},[190],"height:0.6444em;",[19,1384,158],{"className":1385},[195],". Our goal now is to transform the recurrence formula of the Fibonacci sequence into matrix form. How can we do that? We can approach it from the perspective of a system of linear equations.",[10,1388,1389],{},"First, here is what we know:",[19,1391,1393],{"className":1392,"translate":23},[22],[19,1394,1396,1442],{"className":1395,"translate":23},[27],[19,1397,1399],{"className":1398},[31],[33,1400,1401],{"xmlns":35,"display":36},[38,1402,1403,1439],{},[41,1404,1405,1417,1419,1431,1433],{},[44,1406,1407,1409],{},[47,1408,49],{},[41,1410,1411,1413,1415],{},[47,1412,52],{},[54,1414,140],{},[81,1416,107],{},[54,1418,145],{},[44,1420,1421,1423],{},[47,1422,49],{},[41,1424,1425,1427,1429],{},[47,1426,52],{},[54,1428,140],{},[81,1430,158],{},[54,1432,56],{},[44,1434,1435,1437],{},[47,1436,49],{},[47,1438,52],{},[175,1440,1441],{"encoding":177},"F_{n-1} + F_{n-2} = F_{n}",[19,1443,1445,1510,1574],{"className":1444,"ariaHidden":61},[182],[19,1446,1448,1452,1501,1504,1507],{"className":1447},[186],[19,1449],{"className":1450,"style":1451},[190],"height:0.8917em;vertical-align:-0.2083em;",[19,1453,1455,1458],{"className":1454},[195],[19,1456,49],{"className":1457,"style":200},[195,199],[19,1459,1461],{"className":1460},[204],[19,1462,1464,1493],{"className":1463},[208,209],[19,1465,1467,1490],{"className":1466},[213],[19,1468,1470],{"className":1469,"style":450},[217],[19,1471,1472,1475],{"style":221},[19,1473],{"className":1474,"style":226},[225],[19,1476,1478],{"className":1477},[230,231,232,233],[19,1479,1481,1484,1487],{"className":1480},[195,233],[19,1482,52],{"className":1483},[195,199,233],[19,1485,140],{"className":1486},[468,233],[19,1488,107],{"className":1489},[195,233],[19,1491,241],{"className":1492},[240],[19,1494,1496],{"className":1495},[213],[19,1497,1499],{"className":1498,"style":481},[217],[19,1500],{},[19,1502],{"className":1503,"style":487},[254],[19,1505,145],{"className":1506},[468],[19,1508],{"className":1509,"style":487},[254],[19,1511,1513,1516,1565,1568,1571],{"className":1512},[186],[19,1514],{"className":1515,"style":1451},[190],[19,1517,1519,1522],{"className":1518},[195],[19,1520,49],{"className":1521,"style":200},[195,199],[19,1523,1525],{"className":1524},[204],[19,1526,1528,1557],{"className":1527},[208,209],[19,1529,1531,1554],{"className":1530},[213],[19,1532,1534],{"className":1533,"style":450},[217],[19,1535,1536,1539],{"style":221},[19,1537],{"className":1538,"style":226},[225],[19,1540,1542],{"className":1541},[230,231,232,233],[19,1543,1545,1548,1551],{"className":1544},[195,233],[19,1546,52],{"className":1547},[195,199,233],[19,1549,140],{"className":1550},[468,233],[19,1552,158],{"className":1553},[195,233],[19,1555,241],{"className":1556},[240],[19,1558,1560],{"className":1559},[213],[19,1561,1563],{"className":1562,"style":481},[217],[19,1564],{},[19,1566],{"className":1567,"style":255},[254],[19,1569,56],{"className":1570},[259],[19,1572],{"className":1573,"style":255},[254],[19,1575,1577,1580],{"className":1576},[186],[19,1578],{"className":1579,"style":191},[190],[19,1581,1583,1586],{"className":1582},[195],[19,1584,49],{"className":1585,"style":200},[195,199],[19,1587,1589],{"className":1588},[204],[19,1590,1592,1615],{"className":1591},[208,209],[19,1593,1595,1612],{"className":1594},[213],[19,1596,1598],{"className":1597,"style":218},[217],[19,1599,1600,1603],{"style":221},[19,1601],{"className":1602,"style":226},[225],[19,1604,1606],{"className":1605},[230,231,232,233],[19,1607,1609],{"className":1608},[195,233],[19,1610,52],{"className":1611},[195,199,233],[19,1613,241],{"className":1614},[240],[19,1616,1618],{"className":1617},[213],[19,1619,1621],{"className":1620,"style":248},[217],[19,1622],{},[10,1624,1625,1626,1890],{},"We can add the equation ",[19,1627,1629,1686],{"className":1628,"translate":23},[27],[19,1630,1632],{"className":1631},[31],[33,1633,1634],{"xmlns":35},[38,1635,1636,1683],{},[41,1637,1638,1650,1652,1654,1657,1669,1671],{},[44,1639,1640,1642],{},[47,1641,49],{},[41,1643,1644,1646,1648],{},[47,1645,52],{},[54,1647,140],{},[81,1649,107],{},[54,1651,145],{},[81,1653,78],{},[54,1655,1656],{},"⋅",[44,1658,1659,1661],{},[47,1660,49],{},[41,1662,1663,1665,1667],{},[47,1664,52],{},[54,1666,140],{},[81,1668,158],{},[54,1670,56],{},[44,1672,1673,1675],{},[47,1674,49],{},[41,1676,1677,1679,1681],{},[47,1678,52],{},[54,1680,140],{},[81,1682,107],{},[175,1684,1685],{"encoding":177},"F_{n-1} + 0 \\cdot F_{n-2} = F_{n-1}",[19,1687,1689,1753,1771,1835],{"className":1688,"ariaHidden":61},[182],[19,1690,1692,1695,1744,1747,1750],{"className":1691},[186],[19,1693],{"className":1694,"style":1451},[190],[19,1696,1698,1701],{"className":1697},[195],[19,1699,49],{"className":1700,"style":200},[195,199],[19,1702,1704],{"className":1703},[204],[19,1705,1707,1736],{"className":1706},[208,209],[19,1708,1710,1733],{"className":1709},[213],[19,1711,1713],{"className":1712,"style":450},[217],[19,1714,1715,1718],{"style":221},[19,1716],{"className":1717,"style":226},[225],[19,1719,1721],{"className":1720},[230,231,232,233],[19,1722,1724,1727,1730],{"className":1723},[195,233],[19,1725,52],{"className":1726},[195,199,233],[19,1728,140],{"className":1729},[468,233],[19,1731,107],{"className":1732},[195,233],[19,1734,241],{"className":1735},[240],[19,1737,1739],{"className":1738},[213],[19,1740,1742],{"className":1741,"style":481},[217],[19,1743],{},[19,1745],{"className":1746,"style":487},[254],[19,1748,145],{"className":1749},[468],[19,1751],{"className":1752,"style":487},[254],[19,1754,1756,1759,1762,1765,1768],{"className":1755},[186],[19,1757],{"className":1758,"style":1382},[190],[19,1760,78],{"className":1761},[195],[19,1763],{"className":1764,"style":487},[254],[19,1766,1656],{"className":1767},[468],[19,1769],{"className":1770,"style":487},[254],[19,1772,1774,1777,1826,1829,1832],{"className":1773},[186],[19,1775],{"className":1776,"style":1451},[190],[19,1778,1780,1783],{"className":1779},[195],[19,1781,49],{"className":1782,"style":200},[195,199],[19,1784,1786],{"className":1785},[204],[19,1787,1789,1818],{"className":1788},[208,209],[19,1790,1792,1815],{"className":1791},[213],[19,1793,1795],{"className":1794,"style":450},[217],[19,1796,1797,1800],{"style":221},[19,1798],{"className":1799,"style":226},[225],[19,1801,1803],{"className":1802},[230,231,232,233],[19,1804,1806,1809,1812],{"className":1805},[195,233],[19,1807,52],{"className":1808},[195,199,233],[19,1810,140],{"className":1811},[468,233],[19,1813,158],{"className":1814},[195,233],[19,1816,241],{"className":1817},[240],[19,1819,1821],{"className":1820},[213],[19,1822,1824],{"className":1823,"style":481},[217],[19,1825],{},[19,1827],{"className":1828,"style":255},[254],[19,1830,56],{"className":1831},[259],[19,1833],{"className":1834,"style":255},[254],[19,1836,1838,1841],{"className":1837},[186],[19,1839],{"className":1840,"style":1451},[190],[19,1842,1844,1847],{"className":1843},[195],[19,1845,49],{"className":1846,"style":200},[195,199],[19,1848,1850],{"className":1849},[204],[19,1851,1853,1882],{"className":1852},[208,209],[19,1854,1856,1879],{"className":1855},[213],[19,1857,1859],{"className":1858,"style":450},[217],[19,1860,1861,1864],{"style":221},[19,1862],{"className":1863,"style":226},[225],[19,1865,1867],{"className":1866},[230,231,232,233],[19,1868,1870,1873,1876],{"className":1869},[195,233],[19,1871,52],{"className":1872},[195,199,233],[19,1874,140],{"className":1875},[468,233],[19,1877,107],{"className":1878},[195,233],[19,1880,241],{"className":1881},[240],[19,1883,1885],{"className":1884},[213],[19,1886,1888],{"className":1887,"style":481},[217],[19,1889],{}," to form a system of linear equations:",[19,1892,1894],{"className":1893,"translate":23},[22],[19,1895,1897,2007],{"className":1896,"translate":23},[27],[19,1898,1900],{"className":1899},[31],[33,1901,1902],{"xmlns":35,"display":36},[38,1903,1904,2004],{},[41,1905,1906,1908],{},[54,1907,62],{"fence":61},[64,1909,1910,1952],{"rowspacing":66,"columnalign":67,"columnspacing":68},[70,1911,1912],{},[73,1913,1914],{},[76,1915,1916],{"scriptlevel":78,"displaystyle":79},[41,1917,1918,1930,1932,1944,1946],{},[44,1919,1920,1922],{},[47,1921,49],{},[41,1923,1924,1926,1928],{},[47,1925,52],{},[54,1927,140],{},[81,1929,107],{},[54,1931,145],{},[44,1933,1934,1936],{},[47,1935,49],{},[41,1937,1938,1940,1942],{},[47,1939,52],{},[54,1941,140],{},[81,1943,158],{},[54,1945,56],{},[44,1947,1948,1950],{},[47,1949,49],{},[47,1951,52],{},[70,1953,1954],{},[73,1955,1956],{},[76,1957,1958],{"scriptlevel":78,"displaystyle":79},[41,1959,1960,1972,1974,1976,1978,1990,1992],{},[44,1961,1962,1964],{},[47,1963,49],{},[41,1965,1966,1968,1970],{},[47,1967,52],{},[54,1969,140],{},[81,1971,107],{},[54,1973,145],{},[81,1975,78],{},[54,1977,1656],{},[44,1979,1980,1982],{},[47,1981,49],{},[41,1983,1984,1986,1988],{},[47,1985,52],{},[54,1987,140],{},[81,1989,158],{},[54,1991,56],{},[44,1993,1994,1996],{},[47,1995,49],{},[41,1997,1998,2000,2002],{},[47,1999,52],{},[54,2001,140],{},[81,2003,107],{},[175,2005,2006],{"encoding":177},"\\begin{cases}\nF_{n-1} + F_{n-2} = F_{n} \\\\\nF_{n-1} + 0 \\cdot F_{n-2} = F_{n-1}\n\\end{cases}",[19,2008,2010],{"className":2009,"ariaHidden":61},[182],[19,2011,2013,2017],{"className":2012},[186],[19,2014],{"className":2015,"style":2016},[190],"height:3em;vertical-align:-1.25em;",[19,2018,2020,2026,2411],{"className":2019},[273],[19,2021,2023],{"className":2022,"style":979},[277,978],[19,2024,62],{"className":2025},[281,983],[19,2027,2029],{"className":2028},[195],[19,2030,2032],{"className":2031},[64],[19,2033,2035],{"className":2034},[387],[19,2036,2038,2402],{"className":2037},[208,209],[19,2039,2041,2399],{"className":2040},[213],[19,2042,2045,2213],{"className":2043,"style":2044},[217],"height:1.69em;",[19,2046,2048,2051],{"style":2047},"top:-3.69em;",[19,2049],{"className":2050,"style":404},[225],[19,2052,2054,2103,2106,2109,2112,2161,2164,2167,2170],{"className":2053},[195],[19,2055,2057,2060],{"className":2056},[195],[19,2058,49],{"className":2059,"style":200},[195,199],[19,2061,2063],{"className":2062},[204],[19,2064,2066,2095],{"className":2065},[208,209],[19,2067,2069,2092],{"className":2068},[213],[19,2070,2072],{"className":2071,"style":450},[217],[19,2073,2074,2077],{"style":221},[19,2075],{"className":2076,"style":226},[225],[19,2078,2080],{"className":2079},[230,231,232,233],[19,2081,2083,2086,2089],{"className":2082},[195,233],[19,2084,52],{"className":2085},[195,199,233],[19,2087,140],{"className":2088},[468,233],[19,2090,107],{"className":2091},[195,233],[19,2093,241],{"className":2094},[240],[19,2096,2098],{"className":2097},[213],[19,2099,2101],{"className":2100,"style":481},[217],[19,2102],{},[19,2104],{"className":2105,"style":487},[254],[19,2107,145],{"className":2108},[468],[19,2110],{"className":2111,"style":487},[254],[19,2113,2115,2118],{"className":2114},[195],[19,2116,49],{"className":2117,"style":200},[195,199],[19,2119,2121],{"className":2120},[204],[19,2122,2124,2153],{"className":2123},[208,209],[19,2125,2127,2150],{"className":2126},[213],[19,2128,2130],{"className":2129,"style":450},[217],[19,2131,2132,2135],{"style":221},[19,2133],{"className":2134,"style":226},[225],[19,2136,2138],{"className":2137},[230,231,232,233],[19,2139,2141,2144,2147],{"className":2140},[195,233],[19,2142,52],{"className":2143},[195,199,233],[19,2145,140],{"className":2146},[468,233],[19,2148,158],{"className":2149},[195,233],[19,2151,241],{"className":2152},[240],[19,2154,2156],{"className":2155},[213],[19,2157,2159],{"className":2158,"style":481},[217],[19,2160],{},[19,2162],{"className":2163,"style":255},[254],[19,2165,56],{"className":2166},[259],[19,2168],{"className":2169,"style":255},[254],[19,2171,2173,2176],{"className":2172},[195],[19,2174,49],{"className":2175,"style":200},[195,199],[19,2177,2179],{"className":2178},[204],[19,2180,2182,2205],{"className":2181},[208,209],[19,2183,2185,2202],{"className":2184},[213],[19,2186,2188],{"className":2187,"style":218},[217],[19,2189,2190,2193],{"style":221},[19,2191],{"className":2192,"style":226},[225],[19,2194,2196],{"className":2195},[230,231,232,233],[19,2197,2199],{"className":2198},[195,233],[19,2200,52],{"className":2201},[195,199,233],[19,2203,241],{"className":2204},[240],[19,2206,2208],{"className":2207},[213],[19,2209,2211],{"className":2210,"style":248},[217],[19,2212],{},[19,2214,2216,2219],{"style":2215},"top:-2.25em;",[19,2217],{"className":2218,"style":404},[225],[19,2220,2222,2271,2274,2277,2280,2283,2286,2289,2292,2341,2344,2347,2350],{"className":2221},[195],[19,2223,2225,2228],{"className":2224},[195],[19,2226,49],{"className":2227,"style":200},[195,199],[19,2229,2231],{"className":2230},[204],[19,2232,2234,2263],{"className":2233},[208,209],[19,2235,2237,2260],{"className":2236},[213],[19,2238,2240],{"className":2239,"style":450},[217],[19,2241,2242,2245],{"style":221},[19,2243],{"className":2244,"style":226},[225],[19,2246,2248],{"className":2247},[230,231,232,233],[19,2249,2251,2254,2257],{"className":2250},[195,233],[19,2252,52],{"className":2253},[195,199,233],[19,2255,140],{"className":2256},[468,233],[19,2258,107],{"className":2259},[195,233],[19,2261,241],{"className":2262},[240],[19,2264,2266],{"className":2265},[213],[19,2267,2269],{"className":2268,"style":481},[217],[19,2270],{},[19,2272],{"className":2273,"style":487},[254],[19,2275,145],{"className":2276},[468],[19,2278],{"className":2279,"style":487},[254],[19,2281,78],{"className":2282},[195],[19,2284],{"className":2285,"style":487},[254],[19,2287,1656],{"className":2288},[468],[19,2290],{"className":2291,"style":487},[254],[19,2293,2295,2298],{"className":2294},[195],[19,2296,49],{"className":2297,"style":200},[195,199],[19,2299,2301],{"className":2300},[204],[19,2302,2304,2333],{"className":2303},[208,209],[19,2305,2307,2330],{"className":2306},[213],[19,2308,2310],{"className":2309,"style":450},[217],[19,2311,2312,2315],{"style":221},[19,2313],{"className":2314,"style":226},[225],[19,2316,2318],{"className":2317},[230,231,232,233],[19,2319,2321,2324,2327],{"className":2320},[195,233],[19,2322,52],{"className":2323},[195,199,233],[19,2325,140],{"className":2326},[468,233],[19,2328,158],{"className":2329},[195,233],[19,2331,241],{"className":2332},[240],[19,2334,2336],{"className":2335},[213],[19,2337,2339],{"className":2338,"style":481},[217],[19,2340],{},[19,2342],{"className":2343,"style":255},[254],[19,2345,56],{"className":2346},[259],[19,2348],{"className":2349,"style":255},[254],[19,2351,2353,2356],{"className":2352},[195],[19,2354,49],{"className":2355,"style":200},[195,199],[19,2357,2359],{"className":2358},[204],[19,2360,2362,2391],{"className":2361},[208,209],[19,2363,2365,2388],{"className":2364},[213],[19,2366,2368],{"className":2367,"style":450},[217],[19,2369,2370,2373],{"style":221},[19,2371],{"className":2372,"style":226},[225],[19,2374,2376],{"className":2375},[230,231,232,233],[19,2377,2379,2382,2385],{"className":2378},[195,233],[19,2380,52],{"className":2381},[195,199,233],[19,2383,140],{"className":2384},[468,233],[19,2386,107],{"className":2387},[195,233],[19,2389,241],{"className":2390},[240],[19,2392,2394],{"className":2393},[213],[19,2395,2397],{"className":2396,"style":481},[217],[19,2398],{},[19,2400,241],{"className":2401},[240],[19,2403,2405],{"className":2404},[213],[19,2406,2409],{"className":2407,"style":2408},[217],"height:1.19em;",[19,2410],{},[19,2412],{"className":2413},[674,675],[10,2415,2416],{},"This can be transformed into matrix form as follows:",[19,2418,2420],{"className":2419,"translate":23},[22],[19,2421,2423,2560],{"className":2422,"translate":23},[27],[19,2424,2426],{"className":2425},[31],[33,2427,2428],{"xmlns":35,"display":36},[38,2429,2430,2557],{},[41,2431,2432,2472,2517,2519],{},[41,2433,2434,2437,2469],{},[54,2435,2436],{"fence":61},"[",[64,2438,2441,2455],{"rowspacing":2439,"columnalign":2440,"columnspacing":68},"0.16em","center center",[70,2442,2443,2449],{},[73,2444,2445],{},[76,2446,2447],{"scriptlevel":78,"displaystyle":79},[81,2448,107],{},[73,2450,2451],{},[76,2452,2453],{"scriptlevel":78,"displaystyle":79},[81,2454,107],{},[70,2456,2457,2463],{},[73,2458,2459],{},[76,2460,2461],{"scriptlevel":78,"displaystyle":79},[81,2462,107],{},[73,2464,2465],{},[76,2466,2467],{"scriptlevel":78,"displaystyle":79},[81,2468,78],{},[54,2470,2471],{"fence":61},"]",[41,2473,2474,2476,2515],{},[54,2475,2436],{"fence":61},[64,2477,2479,2497],{"rowspacing":2439,"columnalign":2478,"columnspacing":68},"center",[70,2480,2481],{},[73,2482,2483],{},[76,2484,2485],{"scriptlevel":78,"displaystyle":79},[44,2486,2487,2489],{},[47,2488,49],{},[41,2490,2491,2493,2495],{},[47,2492,52],{},[54,2494,140],{},[81,2496,107],{},[70,2498,2499],{},[73,2500,2501],{},[76,2502,2503],{"scriptlevel":78,"displaystyle":79},[44,2504,2505,2507],{},[47,2506,49],{},[41,2508,2509,2511,2513],{},[47,2510,52],{},[54,2512,140],{},[81,2514,158],{},[54,2516,2471],{"fence":61},[54,2518,56],{},[41,2520,2521,2523,2555],{},[54,2522,2436],{"fence":61},[64,2524,2525,2537],{"rowspacing":2439,"columnalign":2478,"columnspacing":68},[70,2526,2527],{},[73,2528,2529],{},[76,2530,2531],{"scriptlevel":78,"displaystyle":79},[44,2532,2533,2535],{},[47,2534,49],{},[47,2536,52],{},[70,2538,2539],{},[73,2540,2541],{},[76,2542,2543],{"scriptlevel":78,"displaystyle":79},[44,2544,2545,2547],{},[47,2546,49],{},[41,2548,2549,2551,2553],{},[47,2550,52],{},[54,2552,140],{},[81,2554,107],{},[54,2556,2471],{"fence":61},[175,2558,2559],{"encoding":177},"\\begin{bmatrix}\n1 & 1 \\\\\n1 & 0\n\\end{bmatrix}\n\\begin{bmatrix}\nF_{n-1} \\\\\nF_{n-2}\n\\end{bmatrix}\n=\n\\begin{bmatrix}\nF_{n} \\\\\nF_{n-1}\n\\end{bmatrix}",[19,2561,2563,2863],{"className":2562,"ariaHidden":61},[182],[19,2564,2566,2570,2693,2696,2854,2857,2860],{"className":2565},[186],[19,2567],{"className":2568,"style":2569},[190],"height:2.4em;vertical-align:-0.95em;",[19,2571,2573,2579,2687],{"className":2572},[273],[19,2574,2576],{"className":2575,"style":979},[277,978],[19,2577,2436],{"className":2578},[281,232],[19,2580,2582],{"className":2581},[195],[19,2583,2585,2635,2639,2642],{"className":2584},[64],[19,2586,2589],{"className":2587},[2588],"col-align-c",[19,2590,2592,2626],{"className":2591},[208,209],[19,2593,2595,2623],{"className":2594},[213],[19,2596,2599,2611],{"className":2597,"style":2598},[217],"height:1.45em;",[19,2600,2602,2605],{"style":2601},"top:-3.61em;",[19,2603],{"className":2604,"style":865},[225],[19,2606,2608],{"className":2607},[195],[19,2609,107],{"className":2610},[195],[19,2612,2614,2617],{"style":2613},"top:-2.41em;",[19,2615],{"className":2616,"style":865},[225],[19,2618,2620],{"className":2619},[195],[19,2621,107],{"className":2622},[195],[19,2624,241],{"className":2625},[240],[19,2627,2629],{"className":2628},[213],[19,2630,2633],{"className":2631,"style":2632},[217],"height:0.95em;",[19,2634],{},[19,2636],{"className":2637,"style":2638},[558],"width:0.5em;",[19,2640],{"className":2641,"style":2638},[558],[19,2643,2645],{"className":2644},[2588],[19,2646,2648,2679],{"className":2647},[208,209],[19,2649,2651,2676],{"className":2650},[213],[19,2652,2654,2665],{"className":2653,"style":2598},[217],[19,2655,2656,2659],{"style":2601},[19,2657],{"className":2658,"style":865},[225],[19,2660,2662],{"className":2661},[195],[19,2663,107],{"className":2664},[195],[19,2666,2667,2670],{"style":2613},[19,2668],{"className":2669,"style":865},[225],[19,2671,2673],{"className":2672},[195],[19,2674,78],{"className":2675},[195],[19,2677,241],{"className":2678},[240],[19,2680,2682],{"className":2681},[213],[19,2683,2685],{"className":2684,"style":2632},[217],[19,2686],{},[19,2688,2690],{"className":2689,"style":979},[674,978],[19,2691,2471],{"className":2692},[281,232],[19,2694],{"className":2695,"style":971},[254],[19,2697,2699,2705,2848],{"className":2698},[273],[19,2700,2702],{"className":2701,"style":979},[277,978],[19,2703,2436],{"className":2704},[281,232],[19,2706,2708],{"className":2707},[195],[19,2709,2711],{"className":2710},[64],[19,2712,2714],{"className":2713},[2588],[19,2715,2717,2840],{"className":2716},[208,209],[19,2718,2720,2837],{"className":2719},[213],[19,2721,2723,2780],{"className":2722,"style":2598},[217],[19,2724,2725,2728],{"style":2601},[19,2726],{"className":2727,"style":865},[225],[19,2729,2731],{"className":2730},[195],[19,2732,2734,2737],{"className":2733},[195],[19,2735,49],{"className":2736,"style":200},[195,199],[19,2738,2740],{"className":2739},[204],[19,2741,2743,2772],{"className":2742},[208,209],[19,2744,2746,2769],{"className":2745},[213],[19,2747,2749],{"className":2748,"style":450},[217],[19,2750,2751,2754],{"style":221},[19,2752],{"className":2753,"style":226},[225],[19,2755,2757],{"className":2756},[230,231,232,233],[19,2758,2760,2763,2766],{"className":2759},[195,233],[19,2761,52],{"className":2762},[195,199,233],[19,2764,140],{"className":2765},[468,233],[19,2767,107],{"className":2768},[195,233],[19,2770,241],{"className":2771},[240],[19,2773,2775],{"className":2774},[213],[19,2776,2778],{"className":2777,"style":481},[217],[19,2779],{},[19,2781,2782,2785],{"style":2613},[19,2783],{"className":2784,"style":865},[225],[19,2786,2788],{"className":2787},[195],[19,2789,2791,2794],{"className":2790},[195],[19,2792,49],{"className":2793,"style":200},[195,199],[19,2795,2797],{"className":2796},[204],[19,2798,2800,2829],{"className":2799},[208,209],[19,2801,2803,2826],{"className":2802},[213],[19,2804,2806],{"className":2805,"style":450},[217],[19,2807,2808,2811],{"style":221},[19,2809],{"className":2810,"style":226},[225],[19,2812,2814],{"className":2813},[230,231,232,233],[19,2815,2817,2820,2823],{"className":2816},[195,233],[19,2818,52],{"className":2819},[195,199,233],[19,2821,140],{"className":2822},[468,233],[19,2824,158],{"className":2825},[195,233],[19,2827,241],{"className":2828},[240],[19,2830,2832],{"className":2831},[213],[19,2833,2835],{"className":2834,"style":481},[217],[19,2836],{},[19,2838,241],{"className":2839},[240],[19,2841,2843],{"className":2842},[213],[19,2844,2846],{"className":2845,"style":2632},[217],[19,2847],{},[19,2849,2851],{"className":2850,"style":979},[674,978],[19,2852,2471],{"className":2853},[281,232],[19,2855],{"className":2856,"style":255},[254],[19,2858,56],{"className":2859},[259],[19,2861],{"className":2862,"style":255},[254],[19,2864,2866,2869],{"className":2865},[186],[19,2867],{"className":2868,"style":2569},[190],[19,2870,2872,2878,3015],{"className":2871},[273],[19,2873,2875],{"className":2874,"style":979},[277,978],[19,2876,2436],{"className":2877},[281,232],[19,2879,2881],{"className":2880},[195],[19,2882,2884],{"className":2883},[64],[19,2885,2887],{"className":2886},[2588],[19,2888,2890,3007],{"className":2889},[208,209],[19,2891,2893,3004],{"className":2892},[213],[19,2894,2896,2947],{"className":2895,"style":2598},[217],[19,2897,2898,2901],{"style":2601},[19,2899],{"className":2900,"style":865},[225],[19,2902,2904],{"className":2903},[195],[19,2905,2907,2910],{"className":2906},[195],[19,2908,49],{"className":2909,"style":200},[195,199],[19,2911,2913],{"className":2912},[204],[19,2914,2916,2939],{"className":2915},[208,209],[19,2917,2919,2936],{"className":2918},[213],[19,2920,2922],{"className":2921,"style":218},[217],[19,2923,2924,2927],{"style":221},[19,2925],{"className":2926,"style":226},[225],[19,2928,2930],{"className":2929},[230,231,232,233],[19,2931,2933],{"className":2932},[195,233],[19,2934,52],{"className":2935},[195,199,233],[19,2937,241],{"className":2938},[240],[19,2940,2942],{"className":2941},[213],[19,2943,2945],{"className":2944,"style":248},[217],[19,2946],{},[19,2948,2949,2952],{"style":2613},[19,2950],{"className":2951,"style":865},[225],[19,2953,2955],{"className":2954},[195],[19,2956,2958,2961],{"className":2957},[195],[19,2959,49],{"className":2960,"style":200},[195,199],[19,2962,2964],{"className":2963},[204],[19,2965,2967,2996],{"className":2966},[208,209],[19,2968,2970,2993],{"className":2969},[213],[19,2971,2973],{"className":2972,"style":450},[217],[19,2974,2975,2978],{"style":221},[19,2976],{"className":2977,"style":226},[225],[19,2979,2981],{"className":2980},[230,231,232,233],[19,2982,2984,2987,2990],{"className":2983},[195,233],[19,2985,52],{"className":2986},[195,199,233],[19,2988,140],{"className":2989},[468,233],[19,2991,107],{"className":2992},[195,233],[19,2994,241],{"className":2995},[240],[19,2997,2999],{"className":2998},[213],[19,3000,3002],{"className":3001,"style":481},[217],[19,3003],{},[19,3005,241],{"className":3006},[240],[19,3008,3010],{"className":3009},[213],[19,3011,3013],{"className":3012,"style":2632},[217],[19,3014],{},[19,3016,3018],{"className":3017,"style":979},[674,978],[19,3019,2471],{"className":3020},[281,232],[10,3022,3023],{},"Now, we can iterate this process:",[19,3025,3027],{"className":3026,"translate":23},[22],[19,3028,3030,3437],{"className":3029,"translate":23},[27],[19,3031,3033],{"className":3032},[31],[33,3034,3035],{"xmlns":35,"display":36},[38,3036,3037,3434],{},[64,3038,3042,3187,3299,3329],{"rowspacing":3039,"columnalign":3040,"columnspacing":3041},"0.25em","right left","0em",[70,3043,3044,3048,3090,3180,3183],{},[73,3045],{"className":3046},[3047],"mtr-glue",[73,3049,3050],{},[76,3051,3052],{"scriptlevel":78,"displaystyle":61},[41,3053,3054,3056,3088],{},[54,3055,2436],{"fence":61},[64,3057,3058,3070],{"rowspacing":2439,"columnalign":2478,"columnspacing":68},[70,3059,3060],{},[73,3061,3062],{},[76,3063,3064],{"scriptlevel":78,"displaystyle":79},[44,3065,3066,3068],{},[47,3067,49],{},[47,3069,52],{},[70,3071,3072],{},[73,3073,3074],{},[76,3075,3076],{"scriptlevel":78,"displaystyle":79},[44,3077,3078,3080],{},[47,3079,49],{},[41,3081,3082,3084,3086],{},[47,3083,52],{},[54,3085,140],{},[81,3087,107],{},[54,3089,2471],{"fence":61},[73,3091,3092],{},[76,3093,3094],{"scriptlevel":78,"displaystyle":61},[41,3095,3096,3098,3100,3136],{},[41,3097],{},[54,3099,56],{},[41,3101,3102,3104,3134],{},[54,3103,2436],{"fence":61},[64,3105,3106,3120],{"rowspacing":2439,"columnalign":2440,"columnspacing":68},[70,3107,3108,3114],{},[73,3109,3110],{},[76,3111,3112],{"scriptlevel":78,"displaystyle":79},[81,3113,107],{},[73,3115,3116],{},[76,3117,3118],{"scriptlevel":78,"displaystyle":79},[81,3119,107],{},[70,3121,3122,3128],{},[73,3123,3124],{},[76,3125,3126],{"scriptlevel":78,"displaystyle":79},[81,3127,107],{},[73,3129,3130],{},[76,3131,3132],{"scriptlevel":78,"displaystyle":79},[81,3133,78],{},[54,3135,2471],{"fence":61},[41,3137,3138,3140,3178],{},[54,3139,2436],{"fence":61},[64,3141,3142,3160],{"rowspacing":2439,"columnalign":2478,"columnspacing":68},[70,3143,3144],{},[73,3145,3146],{},[76,3147,3148],{"scriptlevel":78,"displaystyle":79},[44,3149,3150,3152],{},[47,3151,49],{},[41,3153,3154,3156,3158],{},[47,3155,52],{},[54,3157,140],{},[81,3159,107],{},[70,3161,3162],{},[73,3163,3164],{},[76,3165,3166],{"scriptlevel":78,"displaystyle":79},[44,3167,3168,3170],{},[47,3169,49],{},[41,3171,3172,3174,3176],{},[47,3173,52],{},[54,3175,140],{},[81,3177,158],{},[54,3179,2471],{"fence":61},[73,3181],{"className":3182},[3047],[73,3184],{"className":3185},[3186],"mml-eqn-num",[70,3188,3189,3192,3198,3293,3296],{},[73,3190],{"className":3191},[3047],[73,3193,3194],{},[76,3195,3196],{"scriptlevel":78,"displaystyle":61},[41,3197],{},[73,3199,3200],{},[76,3201,3202],{"scriptlevel":78,"displaystyle":61},[41,3203,3204,3206,3208,3248],{},[41,3205],{},[54,3207,56],{},[719,3209,3210,3246],{},[41,3211,3212,3214,3244],{},[54,3213,2436],{"fence":61},[64,3215,3216,3230],{"rowspacing":2439,"columnalign":2440,"columnspacing":68},[70,3217,3218,3224],{},[73,3219,3220],{},[76,3221,3222],{"scriptlevel":78,"displaystyle":79},[81,3223,107],{},[73,3225,3226],{},[76,3227,3228],{"scriptlevel":78,"displaystyle":79},[81,3229,107],{},[70,3231,3232,3238],{},[73,3233,3234],{},[76,3235,3236],{"scriptlevel":78,"displaystyle":79},[81,3237,107],{},[73,3239,3240],{},[76,3241,3242],{"scriptlevel":78,"displaystyle":79},[81,3243,78],{},[54,3245,2471],{"fence":61},[81,3247,158],{},[41,3249,3250,3252,3291],{},[54,3251,2436],{"fence":61},[64,3253,3254,3272],{"rowspacing":2439,"columnalign":2478,"columnspacing":68},[70,3255,3256],{},[73,3257,3258],{},[76,3259,3260],{"scriptlevel":78,"displaystyle":79},[44,3261,3262,3264],{},[47,3263,49],{},[41,3265,3266,3268,3270],{},[47,3267,52],{},[54,3269,140],{},[81,3271,158],{},[70,3273,3274],{},[73,3275,3276],{},[76,3277,3278],{"scriptlevel":78,"displaystyle":79},[44,3279,3280,3282],{},[47,3281,49],{},[41,3283,3284,3286,3288],{},[47,3285,52],{},[54,3287,140],{},[81,3289,3290],{},"3",[54,3292,2471],{"fence":61},[73,3294],{"className":3295},[3047],[73,3297],{"className":3298},[3186],[70,3300,3301,3304,3310,3323,3326],{},[73,3302],{"className":3303},[3047],[73,3305,3306],{},[76,3307,3308],{"scriptlevel":78,"displaystyle":61},[41,3309],{},[73,3311,3312],{},[76,3313,3314],{"scriptlevel":78,"displaystyle":61},[41,3315,3316,3318,3320],{},[41,3317],{},[54,3319,56],{},[54,3321,3322],{},"⋯",[73,3324],{"className":3325},[3047],[73,3327],{"className":3328},[3186],[70,3330,3331,3334,3340,3428,3431],{},[73,3332],{"className":3333},[3047],[73,3335,3336],{},[76,3337,3338],{"scriptlevel":78,"displaystyle":61},[41,3339],{},[73,3341,3342],{},[76,3343,3344],{"scriptlevel":78,"displaystyle":61},[41,3345,3346,3348,3350,3396],{},[41,3347],{},[54,3349,56],{},[719,3351,3352,3388],{},[41,3353,3354,3356,3386],{},[54,3355,2436],{"fence":61},[64,3357,3358,3372],{"rowspacing":2439,"columnalign":2440,"columnspacing":68},[70,3359,3360,3366],{},[73,3361,3362],{},[76,3363,3364],{"scriptlevel":78,"displaystyle":79},[81,3365,107],{},[73,3367,3368],{},[76,3369,3370],{"scriptlevel":78,"displaystyle":79},[81,3371,107],{},[70,3373,3374,3380],{},[73,3375,3376],{},[76,3377,3378],{"scriptlevel":78,"displaystyle":79},[81,3379,107],{},[73,3381,3382],{},[76,3383,3384],{"scriptlevel":78,"displaystyle":79},[81,3385,78],{},[54,3387,2471],{"fence":61},[41,3389,3390,3392,3394],{},[47,3391,52],{},[54,3393,140],{},[81,3395,107],{},[41,3397,3398,3400,3426],{},[54,3399,2436],{"fence":61},[64,3401,3402,3414],{"rowspacing":2439,"columnalign":2478,"columnspacing":68},[70,3403,3404],{},[73,3405,3406],{},[76,3407,3408],{"scriptlevel":78,"displaystyle":79},[44,3409,3410,3412],{},[47,3411,49],{},[81,3413,107],{},[70,3415,3416],{},[73,3417,3418],{},[76,3419,3420],{"scriptlevel":78,"displaystyle":79},[44,3421,3422,3424],{},[47,3423,49],{},[81,3425,78],{},[54,3427,2471],{"fence":61},[73,3429],{"className":3430},[3047],[73,3432],{"className":3433},[3186],[175,3435,3436],{"encoding":177},"\\begin{align}\n\\begin{bmatrix}\nF_{n} \\\\\nF_{n-1}\n\\end{bmatrix}\n&=\n\\begin{bmatrix}\n1 & 1 \\\\\n1 & 0\n\\end{bmatrix}\n\\begin{bmatrix}\nF_{n-1} \\\\\nF_{n-2}\n\\end{bmatrix} \\\\\n&=\n\\begin{bmatrix}\n1 & 1 \\\\\n1 & 0\n\\end{bmatrix}^2\n\\begin{bmatrix}\nF_{n-2} \\\\\nF_{n-3}\n\\end{bmatrix} \\\\\n&=\\cdots \\\\\n&=\n\\begin{bmatrix}\n1 & 1 \\\\\n1 & 0\n\\end{bmatrix}^{n-1}\n\\begin{bmatrix}\nF_{1} 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denote the matrix ",[19,4717,4719,4773],{"className":4718,"translate":23},[27],[19,4720,4722],{"className":4721},[31],[33,4723,4724],{"xmlns":35},[38,4725,4726,4770],{},[41,4727,4728,4732,4734],{},[47,4729,4731],{"mathvariant":4730},"bold-italic","A",[54,4733,56],{},[41,4735,4736,4738,4768],{},[54,4737,2436],{"fence":61},[64,4739,4740,4754],{"rowspacing":2439,"columnalign":2440,"columnspacing":68},[70,4741,4742,4748],{},[73,4743,4744],{},[76,4745,4746],{"scriptlevel":78,"displaystyle":79},[81,4747,107],{},[73,4749,4750],{},[76,4751,4752],{"scriptlevel":78,"displaystyle":79},[81,4753,107],{},[70,4755,4756,4762],{},[73,4757,4758],{},[76,4759,4760],{"scriptlevel":78,"displaystyle":79},[81,4761,107],{},[73,4763,4764],{},[76,4765,4766],{"scriptlevel":78,"displaystyle":79},[81,4767,78],{},[54,4769,2471],{"fence":61},[175,4771,4772],{"encoding":177},"\\boldsymbol{A} = \\begin{bmatrix}\n1 & 1 \\\\\n1 & 0\n\\end{bmatrix}",[19,4774,4776,4802],{"className":4775,"ariaHidden":61},[182],[19,4777,4779,4783,4793,4796,4799],{"className":4778},[186],[19,4780],{"className":4781,"style":4782},[190],"height:0.6861em;",[19,4784,4786],{"className":4785},[195],[19,4787,4789],{"className":4788},[195],[19,4790,4731],{"className":4791},[195,4792],"boldsymbol",[19,4794],{"className":4795,"style":255},[254],[19,4797,56],{"className":4798},[259],[19,4800],{"className":4801,"style":255},[254],[19,4803,4805,4808],{"className":4804},[186],[19,4806],{"className":4807,"style":2569},[190],[19,4809,4811,4817,4919],{"className":4810},[273],[19,4812,4814],{"className":4813,"style":979},[277,978],[19,4815,2436],{"className":4816},[281,232],[19,4818,4820],{"className":4819},[195],[19,4821,4823,4868,4871,4874],{"className":4822},[64],[19,4824,4826],{"className":4825},[2588],[19,4827,4829,4860],{"className":4828},[208,209],[19,4830,4832,4857],{"className":4831},[213],[19,4833,4835,4846],{"className":4834,"style":2598},[217],[19,4836,4837,4840],{"style":2601},[19,4838],{"className":4839,"style":865},[225],[19,4841,4843],{"className":4842},[195],[19,4844,107],{"className":4845},[195],[19,4847,4848,4851],{"style":2613},[19,4849],{"className":4850,"style":865},[225],[19,4852,4854],{"className":4853},[195],[19,4855,107],{"className":4856},[195],[19,4858,241],{"className":4859},[240],[19,4861,4863],{"className":4862},[213],[19,4864,4866],{"className":4865,"style":2632},[217],[19,4867],{},[19,4869],{"className":4870,"style":2638},[558],[19,4872],{"className":4873,"style":2638},[558],[19,4875,4877],{"className":4876},[2588],[19,4878,4880,4911],{"className":4879},[208,209],[19,4881,4883,4908],{"className":4882},[213],[19,4884,4886,4897],{"className":4885,"style":2598},[217],[19,4887,4888,4891],{"style":2601},[19,4889],{"className":4890,"style":865},[225],[19,4892,4894],{"className":4893},[195],[19,4895,107],{"className":4896},[195],[19,4898,4899,4902],{"style":2613},[19,4900],{"className":4901,"style":865},[225],[19,4903,4905],{"className":4904},[195],[19,4906,78],{"className":4907},[195],[19,4909,241],{"className":4910},[240],[19,4912,4914],{"className":4913},[213],[19,4915,4917],{"className":4916,"style":2632},[217],[19,4918],{},[19,4920,4922],{"className":4921,"style":979},[674,978],[19,4923,2471],{"className":4924},[281,232],". So, the problem becomes finding ",[19,4927,4929,4953],{"className":4928,"translate":23},[27],[19,4930,4932],{"className":4931},[31],[33,4933,4934],{"xmlns":35},[38,4935,4936,4950],{},[41,4937,4938],{},[719,4939,4940,4942],{},[47,4941,4731],{"mathvariant":4730},[41,4943,4944,4946,4948],{},[47,4945,52],{},[54,4947,140],{},[81,4949,107],{},[175,4951,4952],{"encoding":177},"\\boldsymbol{A}^{n-1}",[19,4954,4956],{"className":4955,"ariaHidden":61},[182],[19,4957,4959,4963],{"className":4958},[186],[19,4960],{"className":4961,"style":4962},[190],"height:0.8901em;",[19,4964,4966,4975],{"className":4965},[195],[19,4967,4969],{"className":4968},[195],[19,4970,4972],{"className":4971},[195],[19,4973,4731],{"className":4974},[195,4792],[19,4976,4978],{"className":4977},[204],[19,4979,4981],{"className":4980},[208],[19,4982,4984],{"className":4983},[213],[19,4985,4987],{"className":4986,"style":4962},[217],[19,4988,4990,4993],{"style":4989},"top:-3.139em;margin-right:0.05em;",[19,4991],{"className":4992,"style":226},[225],[19,4994,4996],{"className":4995},[230,231,232,233],[19,4997,4999,5002,5005],{"className":4998},[195,233],[19,5000,52],{"className":5001},[195,199,233],[19,5003,140],{"className":5004},[468,233],[19,5006,107],{"className":5007},[195,233],", and then we can calculate ",[19,5010,5012,5030],{"className":5011,"translate":23},[27],[19,5013,5015],{"className":5014},[31],[33,5016,5017],{"xmlns":35},[38,5018,5019,5027],{},[41,5020,5021],{},[719,5022,5023,5025],{},[47,5024,4731],{"mathvariant":4730},[47,5026,52],{},[175,5028,5029],{"encoding":177},"\\boldsymbol{A}^{n}",[19,5031,5033],{"className":5032,"ariaHidden":61},[182],[19,5034,5036,5040],{"className":5035},[186],[19,5037],{"className":5038,"style":5039},[190],"height:0.7404em;",[19,5041,5043,5052],{"className":5042},[195],[19,5044,5046],{"className":5045},[195],[19,5047,5049],{"className":5048},[195],[19,5050,4731],{"className":5051},[195,4792],[19,5053,5055],{"className":5054},[204],[19,5056,5058],{"className":5057},[208],[19,5059,5061],{"className":5060},[213],[19,5062,5064],{"className":5063,"style":5039},[217],[19,5065,5066,5069],{"style":4989},[19,5067],{"className":5068,"style":226},[225],[19,5070,5072],{"className":5071},[230,231,232,233],[19,5073,5075],{"className":5074},[195,233],[19,5076,52],{"className":5077},[195,199,233]," and replace all instances of ",[19,5080,5082,5095],{"className":5081,"translate":23},[27],[19,5083,5085],{"className":5084},[31],[33,5086,5087],{"xmlns":35},[38,5088,5089,5093],{},[41,5090,5091],{},[47,5092,52],{},[175,5094,52],{"encoding":177},[19,5096,5098],{"className":5097,"ariaHidden":61},[182],[19,5099,5101,5105],{"className":5100},[186],[19,5102],{"className":5103,"style":5104},[190],"height:0.4306em;",[19,5106,52],{"className":5107},[195,199]," with ",[19,5110,5112,5130],{"className":5111,"translate":23},[27],[19,5113,5115],{"className":5114},[31],[33,5116,5117],{"xmlns":35},[38,5118,5119,5127],{},[41,5120,5121,5123,5125],{},[47,5122,52],{},[54,5124,140],{},[81,5126,107],{},[175,5128,5129],{"encoding":177},"n-1",[19,5131,5133,5152],{"className":5132,"ariaHidden":61},[182],[19,5134,5136,5140,5143,5146,5149],{"className":5135},[186],[19,5137],{"className":5138,"style":5139},[190],"height:0.6667em;vertical-align:-0.0833em;",[19,5141,52],{"className":5142},[195,199],[19,5144],{"className":5145,"style":487},[254],[19,5147,140],{"className":5148},[468],[19,5150],{"className":5151,"style":487},[254],[19,5153,5155,5158],{"className":5154},[186],[19,5156],{"className":5157,"style":1382},[190],[19,5159,107],{"className":5160},[195],".",[10,5163,5164,5165,5200],{},"Notice that matrix ",[19,5166,5168,5182],{"className":5167,"translate":23},[27],[19,5169,5171],{"className":5170},[31],[33,5172,5173],{"xmlns":35},[38,5174,5175,5179],{},[41,5176,5177],{},[47,5178,4731],{"mathvariant":4730},[175,5180,5181],{"encoding":177},"\\boldsymbol{A}",[19,5183,5185],{"className":5184,"ariaHidden":61},[182],[19,5186,5188,5191],{"className":5187},[186],[19,5189],{"className":5190,"style":4782},[190],[19,5192,5194],{"className":5193},[195],[19,5195,5197],{"className":5196},[195],[19,5198,4731],{"className":5199},[195,4792]," is a square matrix, and we can utilize the eigenvalues and eigenvectors of matrices.",[10,5202,5203],{},"Eigenvectors can be understood as vectors that, when right-multiplied by the matrix, result in a vector parallel to themselves. Eigenvalues are the scaling factors by which the eigenvectors are scaled when right-multiplied by the matrix. In other words:",[19,5205,5207],{"className":5206,"translate":23},[22],[19,5208,5210,5251],{"className":5209,"translate":23},[27],[19,5211,5213],{"className":5212},[31],[33,5214,5215],{"xmlns":35,"display":36},[38,5216,5217,5248],{},[64,5218,5220],{"width":5219},"100%",[70,5221,5222,5225,5241,5243],{},[73,5223],{"width":5224},"50%",[73,5226,5227],{},[41,5228,5229,5231,5234,5236,5239],{},[47,5230,4731],{"mathvariant":4730},[47,5232,5233],{"mathvariant":4730},"x",[54,5235,56],{},[47,5237,5238],{},"λ",[47,5240,5233],{"mathvariant":4730},[73,5242],{"width":5224},[73,5244,5245],{},[90,5246,5247],{},"(1)",[175,5249,5250],{"encoding":177},"\\boldsymbol{A}\\boldsymbol{x} = \\lambda\\boldsymbol{x}\\tag{1}",[19,5252,5254,5287,5306],{"className":5253,"ariaHidden":61},[182],[19,5255,5257,5260,5269,5278,5281,5284],{"className":5256},[186],[19,5258],{"className":5259,"style":4782},[190],[19,5261,5263],{"className":5262},[195],[19,5264,5266],{"className":5265},[195],[19,5267,4731],{"className":5268},[195,4792],[19,5270,5272],{"className":5271},[195],[19,5273,5275],{"className":5274},[195],[19,5276,5233],{"className":5277},[195,4792],[19,5279],{"className":5280,"style":255},[254],[19,5282,56],{"className":5283},[259],[19,5285],{"className":5286,"style":255},[254],[19,5288,5290,5294,5297],{"className":5289},[186],[19,5291],{"className":5292,"style":5293},[190],"height:0.6944em;",[19,5295,5238],{"className":5296},[195,199],[19,5298,5300],{"className":5299},[195],[19,5301,5303],{"className":5302},[195],[19,5304,5233],{"className":5305},[195,4792],[19,5307,5309,5313],{"className":5308},[4659],[19,5310],{"className":5311,"style":5312},[190],"height:1em;vertical-align:-0.25em;",[19,5314,5316,5319,5325],{"className":5315},[195,583],[19,5317,717],{"className":5318},[195],[19,5320,5322],{"className":5321},[195],[19,5323,107],{"className":5324},[195],[19,5326,741],{"className":5327},[195],[10,5329,5330,5331,5360,5361,5456,5457,5485,5486,5514,5515,5161],{},"Here, ",[19,5332,5334,5348],{"className":5333,"translate":23},[27],[19,5335,5337],{"className":5336},[31],[33,5338,5339],{"xmlns":35},[38,5340,5341,5345],{},[41,5342,5343],{},[47,5344,5238],{},[175,5346,5347],{"encoding":177},"\\lambda",[19,5349,5351],{"className":5350,"ariaHidden":61},[182],[19,5352,5354,5357],{"className":5353},[186],[19,5355],{"className":5356,"style":5293},[190],[19,5358,5238],{"className":5359},[195,199]," is the eigenvalue, and the non-zero vector ",[19,5362,5364,5389],{"className":5363,"translate":23},[27],[19,5365,5367],{"className":5366},[31],[33,5368,5369],{"xmlns":35},[38,5370,5371,5386],{},[41,5372,5373,5375,5378],{},[47,5374,5233],{"mathvariant":4730},[54,5376,5377],{},"∈",[719,5379,5380,5384],{},[47,5381,5383],{"mathvariant":5382},"double-struck","R",[47,5385,52],{},[175,5387,5388],{"encoding":177},"\\boldsymbol{x} \\in \\mathbb{R}^n",[19,5390,5392,5417],{"className":5391,"ariaHidden":61},[182],[19,5393,5395,5399,5408,5411,5414],{"className":5394},[186],[19,5396],{"className":5397,"style":5398},[190],"height:0.5782em;vertical-align:-0.0391em;",[19,5400,5402],{"className":5401},[195],[19,5403,5405],{"className":5404},[195],[19,5406,5233],{"className":5407},[195,4792],[19,5409],{"className":5410,"style":255},[254],[19,5412,5377],{"className":5413},[259],[19,5415],{"className":5416,"style":255},[254],[19,5418,5420,5424],{"className":5419},[186],[19,5421],{"className":5422,"style":5423},[190],"height:0.6889em;",[19,5425,5427,5431],{"className":5426},[195],[19,5428,5383],{"className":5429},[195,5430],"mathbb",[19,5432,5434],{"className":5433},[204],[19,5435,5437],{"className":5436},[208],[19,5438,5440],{"className":5439},[213],[19,5441,5444],{"className":5442,"style":5443},[217],"height:0.6644em;",[19,5445,5447,5450],{"style":5446},"top:-3.063em;margin-right:0.05em;",[19,5448],{"className":5449,"style":226},[225],[19,5451,5453],{"className":5452},[230,231,232,233],[19,5454,52],{"className":5455},[195,199,233]," (where ",[19,5458,5460,5473],{"className":5459,"translate":23},[27],[19,5461,5463],{"className":5462},[31],[33,5464,5465],{"xmlns":35},[38,5466,5467,5471],{},[41,5468,5469],{},[47,5470,52],{},[175,5472,52],{"encoding":177},[19,5474,5476],{"className":5475,"ariaHidden":61},[182],[19,5477,5479,5482],{"className":5478},[186],[19,5480],{"className":5481,"style":5104},[190],[19,5483,52],{"className":5484},[195,199]," is the order of the square matrix) is the eigenvector corresponding to the eigenvalue ",[19,5487,5489,5502],{"className":5488,"translate":23},[27],[19,5490,5492],{"className":5491},[31],[33,5493,5494],{"xmlns":35},[38,5495,5496,5500],{},[41,5497,5498],{},[47,5499,5238],{},[175,5501,5347],{"encoding":177},[19,5503,5505],{"className":5504,"ariaHidden":61},[182],[19,5506,5508,5511],{"className":5507},[186],[19,5509],{"className":5510,"style":5293},[190],[19,5512,5238],{"className":5513},[195,199]," of matrix ",[19,5516,5518,5531],{"className":5517,"translate":23},[27],[19,5519,5521],{"className":5520},[31],[33,5522,5523],{"xmlns":35},[38,5524,5525,5529],{},[41,5526,5527],{},[47,5528,4731],{"mathvariant":4730},[175,5530,5181],{"encoding":177},[19,5532,5534],{"className":5533,"ariaHidden":61},[182],[19,5535,5537,5540],{"className":5536},[186],[19,5538],{"className":5539,"style":4782},[190],[19,5541,5543],{"className":5542},[195],[19,5544,5546],{"className":5545},[195],[19,5547,4731],{"className":5548},[195,4792],[10,5550,5551,5552,5624,5625,5695,5696,5724,5725,5753,5754,5918,5919,5956],{},"So, the specific approach is to first find the eigenvalues ",[19,5553,5555,5573],{"className":5554,"translate":23},[27],[19,5556,5558],{"className":5557},[31],[33,5559,5560],{"xmlns":35},[38,5561,5562,5570],{},[41,5563,5564],{},[44,5565,5566,5568],{},[47,5567,5238],{},[81,5569,107],{},[175,5571,5572],{"encoding":177},"\\lambda_1",[19,5574,5576],{"className":5575,"ariaHidden":61},[182],[19,5577,5579,5583],{"className":5578},[186],[19,5580],{"className":5581,"style":5582},[190],"height:0.8444em;vertical-align:-0.15em;",[19,5584,5586,5589],{"className":5585},[195],[19,5587,5238],{"className":5588},[195,199],[19,5590,5592],{"className":5591},[204],[19,5593,5595,5616],{"className":5594},[208,209],[19,5596,5598,5613],{"className":5597},[213],[19,5599,5601],{"className":5600,"style":450},[217],[19,5602,5604,5607],{"style":5603},"top:-2.55em;margin-left:0em;margin-right:0.05em;",[19,5605],{"className":5606,"style":226},[225],[19,5608,5610],{"className":5609},[230,231,232,233],[19,5611,107],{"className":5612},[195,233],[19,5614,241],{"className":5615},[240],[19,5617,5619],{"className":5618},[213],[19,5620,5622],{"className":5621,"style":248},[217],[19,5623],{}," and ",[19,5626,5628,5646],{"className":5627,"translate":23},[27],[19,5629,5631],{"className":5630},[31],[33,5632,5633],{"xmlns":35},[38,5634,5635,5643],{},[41,5636,5637],{},[44,5638,5639,5641],{},[47,5640,5238],{},[81,5642,158],{},[175,5644,5645],{"encoding":177},"\\lambda_2",[19,5647,5649],{"className":5648,"ariaHidden":61},[182],[19,5650,5652,5655],{"className":5651},[186],[19,5653],{"className":5654,"style":5582},[190],[19,5656,5658,5661],{"className":5657},[195],[19,5659,5238],{"className":5660},[195,199],[19,5662,5664],{"className":5663},[204],[19,5665,5667,5687],{"className":5666},[208,209],[19,5668,5670,5684],{"className":5669},[213],[19,5671,5673],{"className":5672,"style":450},[217],[19,5674,5675,5678],{"style":5603},[19,5676],{"className":5677,"style":226},[225],[19,5679,5681],{"className":5680},[230,231,232,233],[19,5682,158],{"className":5683},[195,233],[19,5685,241],{"className":5686},[240],[19,5688,5690],{"className":5689},[213],[19,5691,5693],{"className":5692,"style":248},[217],[19,5694],{}," (usually, an ",[19,5697,5699,5712],{"className":5698,"translate":23},[27],[19,5700,5702],{"className":5701},[31],[33,5703,5704],{"xmlns":35},[38,5705,5706,5710],{},[41,5707,5708],{},[47,5709,52],{},[175,5711,52],{"encoding":177},[19,5713,5715],{"className":5714,"ariaHidden":61},[182],[19,5716,5718,5721],{"className":5717},[186],[19,5719],{"className":5720,"style":5104},[190],[19,5722,52],{"className":5723},[195,199],"-order matrix has ",[19,5726,5728,5741],{"className":5727,"translate":23},[27],[19,5729,5731],{"className":5730},[31],[33,5732,5733],{"xmlns":35},[38,5734,5735,5739],{},[41,5736,5737],{},[47,5738,52],{},[175,5740,52],{"encoding":177},[19,5742,5744],{"className":5743,"ariaHidden":61},[182],[19,5745,5747,5750],{"className":5746},[186],[19,5748],{"className":5749,"style":5104},[190],[19,5751,52],{"className":5752},[195,199]," eigenvalues) and obtain the diagonal matrix ",[19,5755,5757,5804],{"className":5756,"translate":23},[27],[19,5758,5760],{"className":5759},[31],[33,5761,5762],{"xmlns":35},[38,5763,5764,5801],{},[41,5765,5766,5781,5783,5789,5792,5798],{},[41,5767,5768,5772,5775,5778],{},[47,5769,5771],{"mathvariant":5770},"normal","d",[47,5773,5774],{"mathvariant":5770},"i",[47,5776,5777],{"mathvariant":5770},"a",[47,5779,5780],{"mathvariant":5770},"g",[54,5782,62],{"stretchy":79},[44,5784,5785,5787],{},[47,5786,5238],{},[81,5788,107],{},[54,5790,5791],{"separator":61},",",[44,5793,5794,5796],{},[47,5795,5238],{},[81,5797,158],{},[54,5799,5800],{"stretchy":79},"}",[175,5802,5803],{"encoding":177},"\\rm{diag}\\{\\lambda_1, \\lambda_2\\}",[19,5805,5807],{"className":5806,"ariaHidden":61},[182],[19,5808,5810,5813],{"className":5809},[186],[19,5811],{"className":5812,"style":5312},[190],[19,5814,5816,5825,5828,5868,5872,5875,5915],{"className":5815},[195],[19,5817,5819],{"className":5818},[195],[19,5820,5824],{"className":5821,"style":5823},[195,5822],"mathrm","margin-right:0.01389em;","diag",[19,5826,62],{"className":5827},[277],[19,5829,5831,5834],{"className":5830},[195],[19,5832,5238],{"className":5833},[195,199],[19,5835,5837],{"className":5836},[204],[19,5838,5840,5860],{"className":5839},[208,209],[19,5841,5843,5857],{"className":5842},[213],[19,5844,5846],{"className":5845,"style":450},[217],[19,5847,5848,5851],{"style":5603},[19,5849],{"className":5850,"style":226},[225],[19,5852,5854],{"className":5853},[230,231,232,233],[19,5855,107],{"className":5856},[195,5822,233],[19,5858,241],{"className":5859},[240],[19,5861,5863],{"className":5862},[213],[19,5864,5866],{"className":5865,"style":248},[217],[19,5867],{},[19,5869,5791],{"className":5870},[5871],"mpunct",[19,5873],{"className":5874,"style":971},[254],[19,5876,5878,5881],{"className":5877},[195],[19,5879,5238],{"className":5880},[195,199],[19,5882,5884],{"className":5883},[204],[19,5885,5887,5907],{"className":5886},[208,209],[19,5888,5890,5904],{"className":5889},[213],[19,5891,5893],{"className":5892,"style":450},[217],[19,5894,5895,5898],{"style":5603},[19,5896],{"className":5897,"style":226},[225],[19,5899,5901],{"className":5900},[230,231,232,233],[19,5902,158],{"className":5903},[195,5822,233],[19,5905,241],{"className":5906},[240],[19,5908,5910],{"className":5909},[213],[19,5911,5913],{"className":5912,"style":248},[217],[19,5914],{},[19,5916,5800],{"className":5917},[674],". Then, we find an invertible matrix ",[19,5920,5922,5937],{"className":5921,"translate":23},[27],[19,5923,5925],{"className":5924},[31],[33,5926,5927],{"xmlns":35},[38,5928,5929,5934],{},[41,5930,5931],{},[47,5932,5933],{"mathvariant":4730},"P",[175,5935,5936],{"encoding":177},"\\boldsymbol{P}",[19,5938,5940],{"className":5939,"ariaHidden":61},[182],[19,5941,5943,5946],{"className":5942},[186],[19,5944],{"className":5945,"style":4782},[190],[19,5947,5949],{"className":5948},[195],[19,5950,5952],{"className":5951},[195],[19,5953,5933],{"className":5954,"style":5955},[195,4792],"margin-right:0.15972em;"," such that:",[19,5958,5960],{"className":5959,"translate":23},[22],[19,5961,5963,6021],{"className":5962,"translate":23},[27],[19,5964,5966],{"className":5965},[31],[33,5967,5968],{"xmlns":35,"display":36},[38,5969,5970,6018],{},[41,5971,5972,5982,5984,5986,5988],{},[719,5973,5974,5976],{},[47,5975,5933],{"mathvariant":4730},[41,5977,5978,5980],{},[54,5979,140],{},[81,5981,107],{},[47,5983,4731],{"mathvariant":4730},[47,5985,5933],{"mathvariant":4730},[54,5987,56],{},[41,5989,5990,6000,6002,6008,6010,6016],{},[41,5991,5992,5994,5996,5998],{},[47,5993,5771],{"mathvariant":5770},[47,5995,5774],{"mathvariant":5770},[47,5997,5777],{"mathvariant":5770},[47,5999,5780],{"mathvariant":5770},[54,6001,62],{"stretchy":79},[44,6003,6004,6006],{},[47,6005,5238],{},[81,6007,107],{},[54,6009,5791],{"separator":61},[44,6011,6012,6014],{},[47,6013,5238],{},[81,6015,158],{},[54,6017,5800],{"stretchy":79},[175,6019,6020],{"encoding":177},"\\boldsymbol{P}^{-1}\\boldsymbol{A}\\boldsymbol{P} = \\rm{diag}\\{\\lambda_1, \\lambda_2\\}",[19,6022,6024,6098],{"className":6023,"ariaHidden":61},[182],[19,6025,6027,6030,6071,6080,6089,6092,6095],{"className":6026},[186],[19,6028],{"className":6029,"style":4962},[190],[19,6031,6033,6042],{"className":6032},[195],[19,6034,6036],{"className":6035},[195],[19,6037,6039],{"className":6038},[195],[19,6040,5933],{"className":6041,"style":5955},[195,4792],[19,6043,6045],{"className":6044},[204],[19,6046,6048],{"className":6047},[208],[19,6049,6051],{"className":6050},[213],[19,6052,6054],{"className":6053,"style":4962},[217],[19,6055,6056,6059],{"style":4989},[19,6057],{"className":6058,"style":226},[225],[19,6060,6062],{"className":6061},[230,231,232,233],[19,6063,6065,6068],{"className":6064},[195,233],[19,6066,140],{"className":6067},[195,233],[19,6069,107],{"className":6070},[195,233],[19,6072,6074],{"className":6073},[195],[19,6075,6077],{"className":6076},[195],[19,6078,4731],{"className":6079},[195,4792],[19,6081,6083],{"className":6082},[195],[19,6084,6086],{"className":6085},[195],[19,6087,5933],{"className":6088,"style":5955},[195,4792],[19,6090],{"className":6091,"style":255},[254],[19,6093,56],{"className":6094},[259],[19,6096],{"className":6097,"style":255},[254],[19,6099,6101,6104],{"className":6100},[186],[19,6102],{"className":6103,"style":5312},[190],[19,6105,6107,6113,6116,6156,6159,6162,6202],{"className":6106},[195],[19,6108,6110],{"className":6109},[195],[19,6111,5824],{"className":6112,"style":5823},[195,5822],[19,6114,62],{"className":6115},[277],[19,6117,6119,6122],{"className":6118},[195],[19,6120,5238],{"className":6121},[195,199],[19,6123,6125],{"className":6124},[204],[19,6126,6128,6148],{"className":6127},[208,209],[19,6129,6131,6145],{"className":6130},[213],[19,6132,6134],{"className":6133,"style":450},[217],[19,6135,6136,6139],{"style":5603},[19,6137],{"className":6138,"style":226},[225],[19,6140,6142],{"className":6141},[230,231,232,233],[19,6143,107],{"className":6144},[195,5822,233],[19,6146,241],{"className":6147},[240],[19,6149,6151],{"className":6150},[213],[19,6152,6154],{"className":6153,"style":248},[217],[19,6155],{},[19,6157,5791],{"className":6158},[5871],[19,6160],{"className":6161,"style":971},[254],[19,6163,6165,6168],{"className":6164},[195],[19,6166,5238],{"className":6167},[195,199],[19,6169,6171],{"className":6170},[204],[19,6172,6174,6194],{"className":6173},[208,209],[19,6175,6177,6191],{"className":6176},[213],[19,6178,6180],{"className":6179,"style":450},[217],[19,6181,6182,6185],{"style":5603},[19,6183],{"className":6184,"style":226},[225],[19,6186,6188],{"className":6187},[230,231,232,233],[19,6189,158],{"className":6190},[195,5822,233],[19,6192,241],{"className":6193},[240],[19,6195,6197],{"className":6196},[213],[19,6198,6200],{"className":6199,"style":248},[217],[19,6201],{},[19,6203,5800],{"className":6204},[674],[10,6206,6207],{},"Using matrix multiplication properties:",[19,6209,6211],{"className":6210,"translate":23},[22],[19,6212,6214,6337],{"className":6213,"translate":23},[27],[19,6215,6217],{"className":6216},[31],[33,6218,6219],{"xmlns":35,"display":36},[38,6220,6221,6334],{},[64,6222,6223],{"width":5219},[70,6224,6225,6227,6327,6329],{},[73,6226],{"width":5224},[73,6228,6229],{},[41,6230,6231,6233,6243,6245,6247,6253,6255,6265,6267,6269,6271,6281,6283,6285,6287,6289,6291,6301,6303,6305,6307,6309,6319,6325],{},[54,6232,717],{"stretchy":79},[719,6234,6235,6237],{},[47,6236,5933],{"mathvariant":4730},[41,6238,6239,6241],{},[54,6240,140],{},[81,6242,107],{},[47,6244,4731],{"mathvariant":4730},[47,6246,5933],{"mathvariant":4730},[719,6248,6249,6251],{},[54,6250,741],{"stretchy":79},[47,6252,52],{},[54,6254,56],{},[719,6256,6257,6259],{},[47,6258,5933],{"mathvariant":4730},[41,6260,6261,6263],{},[54,6262,140],{},[81,6264,107],{},[47,6266,4731],{"mathvariant":4730},[54,6268,717],{"stretchy":79},[47,6270,5933],{"mathvariant":4730},[719,6272,6273,6275],{},[47,6274,5933],{"mathvariant":4730},[41,6276,6277,6279],{},[54,6278,140],{},[81,6280,107],{},[54,6282,741],{"stretchy":79},[47,6284,4731],{"mathvariant":4730},[54,6286,717],{"stretchy":79},[47,6288,5933],{"mathvariant":4730},[54,6290,3322],{},[719,6292,6293,6295],{},[47,6294,5933],{"mathvariant":4730},[41,6296,6297,6299],{},[54,6298,140],{},[81,6300,107],{},[54,6302,741],{"stretchy":79},[47,6304,4731],{"mathvariant":4730},[47,6306,5933],{"mathvariant":4730},[54,6308,56],{},[719,6310,6311,6313],{},[47,6312,5933],{"mathvariant":4730},[41,6314,6315,6317],{},[54,6316,140],{},[81,6318,107],{},[719,6320,6321,6323],{},[47,6322,4731],{"mathvariant":4730},[47,6324,52],{},[47,6326,5933],{"mathvariant":4730},[73,6328],{"width":5224},[73,6330,6331],{},[90,6332,6333],{},"(2)",[175,6335,6336],{"encoding":177},"(\\boldsymbol{P}^{-1}\\boldsymbol{A}\\boldsymbol{P})^n = \\boldsymbol{P}^{-1}\\boldsymbol{A}(\\boldsymbol{P}\\boldsymbol{P}^{-1})\\boldsymbol{A}(\\boldsymbol{P}\\cdots\\boldsymbol{P}^{-1})\\boldsymbol{A}\\boldsymbol{P} = \\boldsymbol{P}^{-1}\\boldsymbol{A}^n\\boldsymbol{P}\\tag{2}",[19,6338,6340,6449,6662,6753],{"className":6339,"ariaHidden":61},[182],[19,6341,6343,6347,6350,6391,6400,6409,6440,6443,6446],{"className":6342},[186],[19,6344],{"className":6345,"style":6346},[190],"height:1.1401em;vertical-align:-0.25em;",[19,6348,717],{"className":6349},[277],[19,6351,6353,6362],{"className":6352},[195],[19,6354,6356],{"className":6355},[195],[19,6357,6359],{"className":6358},[195],[19,6360,5933],{"className":6361,"style":5955},[195,4792],[19,6363,6365],{"className":6364},[204],[19,6366,6368],{"className":6367},[208],[19,6369,6371],{"className":6370},[213],[19,6372,6374],{"className":6373,"style":4962},[217],[19,6375,6376,6379],{"style":4989},[19,6377],{"className":6378,"style":226},[225],[19,6380,6382],{"className":6381},[230,231,232,233],[19,6383,6385,6388],{"className":6384},[195,233],[19,6386,140],{"className":6387},[195,233],[19,6389,107],{"className":6390},[195,233],[19,6392,6394],{"className":6393},[195],[19,6395,6397],{"className":6396},[195],[19,6398,4731],{"className":6399},[195,4792],[19,6401,6403],{"className":6402},[195],[19,6404,6406],{"className":6405},[195],[19,6407,5933],{"className":6408,"style":5955},[195,4792],[19,6410,6412,6415],{"className":6411},[674],[19,6413,741],{"className":6414},[674],[19,6416,6418],{"className":6417},[204],[19,6419,6421],{"className":6420},[208],[19,6422,6424],{"className":6423},[213],[19,6425,6428],{"className":6426,"style":6427},[217],"height:0.7144em;",[19,6429,6431,6434],{"style":6430},"top:-3.113em;margin-right:0.05em;",[19,6432],{"className":6433,"style":226},[225],[19,6435,6437],{"className":6436},[230,231,232,233],[19,6438,52],{"className":6439},[195,199,233],[19,6441],{"className":6442,"style":255},[254],[19,6444,56],{"className":6445},[259],[19,6447],{"className":6448,"style":255},[254],[19,6450,6452,6455,6496,6505,6508,6517,6558,6561,6570,6573,6582,6585,6588,6591,6632,6635,6644,6653,6656,6659],{"className":6451},[186],[19,6453],{"className":6454,"style":6346},[190],[19,6456,6458,6467],{"className":6457},[195],[19,6459,6461],{"className":6460},[195],[19,6462,6464],{"className":6463},[195],[19,6465,5933],{"className":6466,"style":5955},[195,4792],[19,6468,6470],{"className":6469},[204],[19,6471,6473],{"className":6472},[208],[19,6474,6476],{"className":6475},[213],[19,6477,6479],{"className":6478,"style":4962},[217],[19,6480,6481,6484],{"style":4989},[19,6482],{"className":6483,"style":226},[225],[19,6485,6487],{"className":6486},[230,231,232,233],[19,6488,6490,6493],{"className":6489},[195,233],[19,6491,140],{"className":6492},[195,233],[19,6494,107],{"className":6495},[195,233],[19,6497,6499],{"className":6498},[195],[19,6500,6502],{"className":6501},[195],[19,6503,4731],{"className":6504},[195,4792],[19,6506,717],{"className":6507},[277],[19,6509,6511],{"className":6510},[195],[19,6512,6514],{"className":6513},[195],[19,6515,5933],{"className":6516,"style":5955},[195,4792],[19,6518,6520,6529],{"className":6519},[195],[19,6521,6523],{"className":6522},[195],[19,6524,6526],{"className":6525},[195],[19,6527,5933],{"className":6528,"style":5955},[195,4792],[19,6530,6532],{"className":6531},[204],[19,6533,6535],{"className":6534},[208],[19,6536,6538],{"className":6537},[213],[19,6539,6541],{"className":6540,"style":4962},[217],[19,6542,6543,6546],{"style":4989},[19,6544],{"className":6545,"style":226},[225],[19,6547,6549],{"className":6548},[230,231,232,233],[19,6550,6552,6555],{"className":6551},[195,233],[19,6553,140],{"className":6554},[195,233],[19,6556,107],{"className":6557},[195,233],[19,6559,741],{"className":6560},[674],[19,6562,6564],{"className":6563},[195],[19,6565,6567],{"className":6566},[195],[19,6568,4731],{"className":6569},[195,4792],[19,6571,717],{"className":6572},[277],[19,6574,6576],{"className":6575},[195],[19,6577,6579],{"className":6578},[195],[19,6580,5933],{"className":6581,"style":5955},[195,4792],[19,6583],{"className":6584,"style":971},[254],[19,6586,3322],{"className":6587},[273],[19,6589],{"className":6590,"style":971},[254],[19,6592,6594,6603],{"className":6593},[195],[19,6595,6597],{"className":6596},[195],[19,6598,6600],{"className":6599},[195],[19,6601,5933],{"className":6602,"style":5955},[195,4792],[19,6604,6606],{"className":6605},[204],[19,6607,6609],{"className":6608},[208],[19,6610,6612],{"className":6611},[213],[19,6613,6615],{"className":6614,"style":4962},[217],[19,6616,6617,6620],{"style":4989},[19,6618],{"className":6619,"style":226},[225],[19,6621,6623],{"className":6622},[230,231,232,233],[19,6624,6626,6629],{"className":6625},[195,233],[19,6627,140],{"className":6628},[195,233],[19,6630,107],{"className":6631},[195,233],[19,6633,741],{"className":6634},[674],[19,6636,6638],{"className":6637},[195],[19,6639,6641],{"className":6640},[195],[19,6642,4731],{"className":6643},[195,4792],[19,6645,6647],{"className":6646},[195],[19,6648,6650],{"className":6649},[195],[19,6651,5933],{"className":6652,"style":5955},[195,4792],[19,6654],{"className":6655,"style":255},[254],[19,6657,56],{"className":6658},[259],[19,6660],{"className":6661,"style":255},[254],[19,6663,6665,6668,6709,6744],{"className":6664},[186],[19,6666],{"className":6667,"style":4962},[190],[19,6669,6671,6680],{"className":6670},[195],[19,6672,6674],{"className":6673},[195],[19,6675,6677],{"className":6676},[195],[19,6678,5933],{"className":6679,"style":5955},[195,4792],[19,6681,6683],{"className":6682},[204],[19,6684,6686],{"className":6685},[208],[19,6687,6689],{"className":6688},[213],[19,6690,6692],{"className":6691,"style":4962},[217],[19,6693,6694,6697],{"style":4989},[19,6695],{"className":6696,"style":226},[225],[19,6698,6700],{"className":6699},[230,231,232,233],[19,6701,6703,6706],{"className":6702},[195,233],[19,6704,140],{"className":6705},[195,233],[19,6707,107],{"className":6708},[195,233],[19,6710,6712,6721],{"className":6711},[195],[19,6713,6715],{"className":6714},[195],[19,6716,6718],{"className":6717},[195],[19,6719,4731],{"className":6720},[195,4792],[19,6722,6724],{"className":6723},[204],[19,6725,6727],{"className":6726},[208],[19,6728,6730],{"className":6729},[213],[19,6731,6733],{"className":6732,"style":5039},[217],[19,6734,6735,6738],{"style":4989},[19,6736],{"className":6737,"style":226},[225],[19,6739,6741],{"className":6740},[230,231,232,233],[19,6742,52],{"className":6743},[195,199,233],[19,6745,6747],{"className":6746},[195],[19,6748,6750],{"className":6749},[195],[19,6751,5933],{"className":6752,"style":5955},[195,4792],[19,6754,6756,6759],{"className":6755},[4659],[19,6757],{"className":6758,"style":6346},[190],[19,6760,6762,6765,6771],{"className":6761},[195,583],[19,6763,717],{"className":6764},[195],[19,6766,6768],{"className":6767},[195],[19,6769,158],{"className":6770},[195],[19,6772,741],{"className":6773},[195],[10,6775,6776,6777,5161],{},"This allows us to calculate ",[19,6778,6780,6797],{"className":6779,"translate":23},[27],[19,6781,6783],{"className":6782},[31],[33,6784,6785],{"xmlns":35},[38,6786,6787,6795],{},[41,6788,6789],{},[719,6790,6791,6793],{},[47,6792,4731],{"mathvariant":4730},[47,6794,52],{},[175,6796,5029],{"encoding":177},[19,6798,6800],{"className":6799,"ariaHidden":61},[182],[19,6801,6803,6806],{"className":6802},[186],[19,6804],{"className":6805,"style":5039},[190],[19,6807,6809,6818],{"className":6808},[195],[19,6810,6812],{"className":6811},[195],[19,6813,6815],{"className":6814},[195],[19,6816,4731],{"className":6817},[195,4792],[19,6819,6821],{"className":6820},[204],[19,6822,6824],{"className":6823},[208],[19,6825,6827],{"className":6826},[213],[19,6828,6830],{"className":6829,"style":5039},[217],[19,6831,6832,6835],{"style":4989},[19,6833],{"className":6834,"style":226},[225],[19,6836,6838],{"className":6837},[230,231,232,233],[19,6839,6841],{"className":6840},[195,233],[19,6842,52],{"className":6843},[195,199,233],[10,6845,6846,6847,6885],{},"Let's first find the eigenvalues. We can rewrite equation ",[19,6848,6850,6867],{"className":6849,"translate":23},[27],[19,6851,6853],{"className":6852},[31],[33,6854,6855],{"xmlns":35},[38,6856,6857,6865],{},[41,6858,6859,6861,6863],{},[54,6860,717],{"stretchy":79},[81,6862,107],{},[54,6864,741],{"stretchy":79},[175,6866,5247],{"encoding":177},[19,6868,6870],{"className":6869,"ariaHidden":61},[182],[19,6871,6873,6876,6879,6882],{"className":6872},[186],[19,6874],{"className":6875,"style":5312},[190],[19,6877,717],{"className":6878},[277],[19,6880,107],{"className":6881},[195],[19,6883,741],{"className":6884},[674]," as:",[19,6887,6889],{"className":6888,"translate":23},[22],[19,6890,6892,6939],{"className":6891,"translate":23},[27],[19,6893,6895],{"className":6894},[31],[33,6896,6897],{"xmlns":35,"display":36},[38,6898,6899,6936],{},[64,6900,6901],{"width":5219},[70,6902,6903,6905,6929,6931],{},[73,6904],{"width":5224},[73,6906,6907],{},[41,6908,6909,6911,6913,6915,6917,6920,6922,6924,6926],{},[54,6910,717],{"stretchy":79},[47,6912,4731],{"mathvariant":4730},[54,6914,140],{},[47,6916,5238],{},[47,6918,6919],{"mathvariant":4730},"E",[54,6921,741],{"stretchy":79},[47,6923,5233],{"mathvariant":4730},[54,6925,56],{},[47,6927,78],{"mathvariant":6928},"bold",[73,6930],{"width":5224},[73,6932,6933],{},[90,6934,6935],{},"(3)",[175,6937,6938],{"encoding":177},"(\\boldsymbol{A}-\\lambda\\boldsymbol{E})\\boldsymbol{x} = \\boldsymbol{0}\\tag{3}",[19,6940,6942,6969,7009,7025],{"className":6941,"ariaHidden":61},[182],[19,6943,6945,6948,6951,6960,6963,6966],{"className":6944},[186],[19,6946],{"className":6947,"style":5312},[190],[19,6949,717],{"className":6950},[277],[19,6952,6954],{"className":6953},[195],[19,6955,6957],{"className":6956},[195],[19,6958,4731],{"className":6959},[195,4792],[19,6961],{"className":6962,"style":487},[254],[19,6964,140],{"className":6965},[468],[19,6967],{"className":6968,"style":487},[254],[19,6970,6972,6975,6978,6988,6991,7000,7003,7006],{"className":6971},[186],[19,6973],{"className":6974,"style":5312},[190],[19,6976,5238],{"className":6977},[195,199],[19,6979,6981],{"className":6980},[195],[19,6982,6984],{"className":6983},[195],[19,6985,6919],{"className":6986,"style":6987},[195,4792],"margin-right:0.05451em;",[19,6989,741],{"className":6990},[674],[19,6992,6994],{"className":6993},[195],[19,6995,6997],{"className":6996},[195],[19,6998,5233],{"className":6999},[195,4792],[19,7001],{"className":7002,"style":255},[254],[19,7004,56],{"className":7005},[259],[19,7007],{"className":7008,"style":255},[254],[19,7010,7012,7015],{"className":7011},[186],[19,7013],{"className":7014,"style":1382},[190],[19,7016,7018],{"className":7017},[195],[19,7019,7021],{"className":7020},[195],[19,7022,78],{"className":7023},[195,7024],"mathbf",[19,7026,7028,7031],{"className":7027},[4659],[19,7029],{"className":7030,"style":5312},[190],[19,7032,7034,7037,7043],{"className":7033},[195,583],[19,7035,717],{"className":7036},[195],[19,7038,7040],{"className":7039},[195],[19,7041,3290],{"className":7042},[195],[19,7044,741],{"className":7045},[195],[10,7047,5330,7048,7083,7084,7348],{},[19,7049,7051,7065],{"className":7050,"translate":23},[27],[19,7052,7054],{"className":7053},[31],[33,7055,7056],{"xmlns":35},[38,7057,7058,7062],{},[41,7059,7060],{},[47,7061,6919],{"mathvariant":4730},[175,7063,7064],{"encoding":177},"\\boldsymbol{E}",[19,7066,7068],{"className":7067,"ariaHidden":61},[182],[19,7069,7071,7074],{"className":7070},[186],[19,7072],{"className":7073,"style":4782},[190],[19,7075,7077],{"className":7076},[195],[19,7078,7080],{"className":7079},[195],[19,7081,6919],{"className":7082,"style":6987},[195,4792]," is the identity matrix, and we can compute that ",[19,7085,7087,7155],{"className":7086,"translate":23},[27],[19,7088,7090],{"className":7089},[31],[33,7091,7092],{"xmlns":35},[38,7093,7094,7152],{},[41,7095,7096,7098,7100,7102,7104,7106],{},[47,7097,4731],{"mathvariant":4730},[54,7099,140],{},[47,7101,5238],{},[47,7103,6919],{"mathvariant":4730},[54,7105,56],{},[41,7107,7108,7110,7150],{},[54,7109,2436],{"fence":61},[64,7111,7112,7132],{"rowspacing":2439,"columnalign":2440,"columnspacing":68},[70,7113,7114,7126],{},[73,7115,7116],{},[76,7117,7118],{"scriptlevel":78,"displaystyle":79},[41,7119,7120,7122,7124],{},[81,7121,107],{},[54,7123,140],{},[47,7125,5238],{},[73,7127,7128],{},[76,7129,7130],{"scriptlevel":78,"displaystyle":79},[81,7131,107],{},[70,7133,7134,7140],{},[73,7135,7136],{},[76,7137,7138],{"scriptlevel":78,"displaystyle":79},[81,7139,107],{},[73,7141,7142],{},[76,7143,7144],{"scriptlevel":78,"displaystyle":79},[41,7145,7146,7148],{},[54,7147,140],{},[47,7149,5238],{},[54,7151,2471],{"fence":61},[175,7153,7154],{"encoding":177},"\\boldsymbol{A}-\\lambda\\boldsymbol{E} = \\begin{bmatrix}\n1-\\lambda & 1 \\\\\n1 & -\\lambda\n\\end{bmatrix}",[19,7156,7158,7183,7210],{"className":7157,"ariaHidden":61},[182],[19,7159,7161,7165,7174,7177,7180],{"className":7160},[186],[19,7162],{"className":7163,"style":7164},[190],"height:0.7694em;vertical-align:-0.0833em;",[19,7166,7168],{"className":7167},[195],[19,7169,7171],{"className":7170},[195],[19,7172,4731],{"className":7173},[195,4792],[19,7175],{"className":7176,"style":487},[254],[19,7178,140],{"className":7179},[468],[19,7181],{"className":7182,"style":487},[254],[19,7184,7186,7189,7192,7201,7204,7207],{"className":7185},[186],[19,7187],{"className":7188,"style":5293},[190],[19,7190,5238],{"className":7191},[195,199],[19,7193,7195],{"className":7194},[195],[19,7196,7198],{"className":7197},[195],[19,7199,6919],{"className":7200,"style":6987},[195,4792],[19,7202],{"className":7203,"style":255},[254],[19,7205,56],{"className":7206},[259],[19,7208],{"className":7209,"style":255},[254],[19,7211,7213,7216],{"className":7212},[186],[19,7214],{"className":7215,"style":2569},[190],[19,7217,7219,7225,7342],{"className":7218},[273],[19,7220,7222],{"className":7221,"style":979},[277,978],[19,7223,2436],{"className":7224},[281,232],[19,7226,7228],{"className":7227},[195],[19,7229,7231,7288,7291,7294],{"className":7230},[64],[19,7232,7234],{"className":7233},[2588],[19,7235,7237,7280],{"className":7236},[208,209],[19,7238,7240,7277],{"className":7239},[213],[19,7241,7243,7266],{"className":7242,"style":2598},[217],[19,7244,7245,7248],{"style":2601},[19,7246],{"className":7247,"style":865},[225],[19,7249,7251,7254,7257,7260,7263],{"className":7250},[195],[19,7252,107],{"className":7253},[195],[19,7255],{"className":7256,"style":487},[254],[19,7258,140],{"className":7259},[468],[19,7261],{"className":7262,"style":487},[254],[19,7264,5238],{"className":7265},[195,199],[19,7267,7268,7271],{"style":2613},[19,7269],{"className":7270,"style":865},[225],[19,7272,7274],{"className":7273},[195],[19,7275,107],{"className":7276},[195],[19,7278,241],{"className":7279},[240],[19,7281,7283],{"className":7282},[213],[19,7284,7286],{"className":7285,"style":2632},[217],[19,7287],{},[19,7289],{"className":7290,"style":2638},[558],[19,7292],{"className":7293,"style":2638},[558],[19,7295,7297],{"className":7296},[2588],[19,7298,7300,7334],{"className":7299},[208,209],[19,7301,7303,7331],{"className":7302},[213],[19,7304,7306,7317],{"className":7305,"style":2598},[217],[19,7307,7308,7311],{"style":2601},[19,7309],{"className":7310,"style":865},[225],[19,7312,7314],{"className":7313},[195],[19,7315,107],{"className":7316},[195],[19,7318,7319,7322],{"style":2613},[19,7320],{"className":7321,"style":865},[225],[19,7323,7325,7328],{"className":7324},[195],[19,7326,140],{"className":7327},[195],[19,7329,5238],{"className":7330},[195,199],[19,7332,241],{"className":7333},[240],[19,7335,7337],{"className":7336},[213],[19,7338,7340],{"className":7339,"style":2632},[217],[19,7341],{},[19,7343,7345],{"className":7344,"style":979},[674,978],[19,7346,2471],{"className":7347},[281,232],". To ensure that there is a non-zero solution, we solve for:",[19,7350,7352],{"className":7351,"translate":23},[22],[19,7353,7355,7450],{"className":7354,"translate":23},[27],[19,7356,7358],{"className":7357},[31],[33,7359,7360],{"xmlns":35,"display":36},[38,7361,7362,7447],{},[41,7363,7364,7379,7381,7427,7429,7435,7437,7439,7441,7443,7445],{},[41,7365,7366,7369,7371,7373,7375,7377],{},[54,7367,7368],{"fence":61},"∣",[47,7370,4731],{"mathvariant":4730},[54,7372,140],{},[47,7374,5238],{},[47,7376,6919],{"mathvariant":4730},[54,7378,7368],{"fence":61},[54,7380,56],{},[41,7382,7383,7385,7425],{},[54,7384,7368],{"fence":61},[64,7386,7387,7407],{"rowspacing":2439,"columnalign":2440,"columnspacing":68},[70,7388,7389,7401],{},[73,7390,7391],{},[76,7392,7393],{"scriptlevel":78,"displaystyle":79},[41,7394,7395,7397,7399],{},[81,7396,107],{},[54,7398,140],{},[47,7400,5238],{},[73,7402,7403],{},[76,7404,7405],{"scriptlevel":78,"displaystyle":79},[81,7406,107],{},[70,7408,7409,7415],{},[73,7410,7411],{},[76,7412,7413],{"scriptlevel":78,"displaystyle":79},[81,7414,107],{},[73,7416,7417],{},[76,7418,7419],{"scriptlevel":78,"displaystyle":79},[41,7420,7421,7423],{},[54,7422,140],{},[47,7424,5238],{},[54,7426,7368],{"fence":61},[54,7428,56],{},[719,7430,7431,7433],{},[47,7432,5238],{},[81,7434,158],{},[54,7436,140],{},[47,7438,5238],{},[54,7440,140],{},[81,7442,107],{},[54,7444,56],{},[81,7446,78],{},[175,7448,7449],{"encoding":177},"\\left| \\boldsymbol{A}-\\lambda\\boldsymbol{E} \\right| = \\begin{vmatrix}\n1-\\lambda & 1 \\\\\n1 & -\\lambda\n\\end{vmatrix} = \\lambda^2 - \\lambda - 1 = 0",[19,7451,7453,7507,7723,7769,7788,7806],{"className":7452,"ariaHidden":61},[182],[19,7454,7456,7459,7498,7501,7504],{"className":7455},[186],[19,7457],{"className":7458,"style":5312},[190],[19,7460,7462,7465,7474,7477,7480,7483,7486,7495],{"className":7461},[273],[19,7463,7368],{"className":7464,"style":979},[277,978],[19,7466,7468],{"className":7467},[195],[19,7469,7471],{"className":7470},[195],[19,7472,4731],{"className":7473},[195,4792],[19,7475],{"className":7476,"style":487},[254],[19,7478,140],{"className":7479},[468],[19,7481],{"className":7482,"style":487},[254],[19,7484,5238],{"className":7485},[195,199],[19,7487,7489],{"className":7488},[195],[19,7490,7492],{"className":7491},[195],[19,7493,6919],{"className":7494,"style":6987},[195,4792],[19,7496,7368],{"className":7497,"style":979},[674,978],[19,7499],{"className":7500,"style":255},[254],[19,7502,56],{"className":7503},[259],[19,7505],{"className":7506,"style":255},[254],[19,7508,7510,7513,7714,7717,7720],{"className":7509},[186],[19,7511],{"className":7512,"style":2569},[190],[19,7514,7516,7560,7677],{"className":7515},[273],[19,7517,7519],{"className":7518},[277],[19,7520,7522],{"className":7521},[281,282],[19,7523,7525,7552],{"className":7524},[208,209],[19,7526,7528,7549],{"className":7527},[213],[19,7529,7531],{"className":7530,"style":2598},[217],[19,7532,7534,7538],{"style":7533},"top:-3.45em;",[19,7535],{"className":7536,"style":7537},[225],"height:4.4em;",[19,7539,7541],{"style":7540},"width:0.333em;height:2.400em;",[318,7542,7546],{"xmlns":320,"width":7543,"height":7544,"viewBox":7545},"0.333em","2.400em","0 0 333 2400",[327,7547],{"d":7548},"M145 15 v585 v1200 v585 c2.667,10,9.667,15,21,15\nc10,0,16.667,-5,20,-15 v-585 v-1200 v-585 c-2.667,-10,-9.667,-15,-21,-15\nc-10,0,-16.667,5,-20,15z M188 15 H145 v585 v1200 v585 h43z",[19,7550,241],{"className":7551},[240],[19,7553,7555],{"className":7554},[213],[19,7556,7558],{"className":7557,"style":2632},[217],[19,7559],{},[19,7561,7563],{"className":7562},[195],[19,7564,7566,7623,7626,7629],{"className":7565},[64],[19,7567,7569],{"className":7568},[2588],[19,7570,7572,7615],{"className":7571},[208,209],[19,7573,7575,7612],{"className":7574},[213],[19,7576,7578,7601],{"className":7577,"style":2598},[217],[19,7579,7580,7583],{"style":2601},[19,7581],{"className":7582,"style":865},[225],[19,7584,7586,7589,7592,7595,7598],{"className":7585},[195],[19,7587,107],{"className":7588},[195],[19,7590],{"className":7591,"style":487},[254],[19,7593,140],{"className":7594},[468],[19,7596],{"className":7597,"style":487},[254],[19,7599,5238],{"className":7600},[195,199],[19,7602,7603,7606],{"style":2613},[19,7604],{"className":7605,"style":865},[225],[19,7607,7609],{"className":7608},[195],[19,7610,107],{"className":7611},[195],[19,7613,241],{"className":7614},[240],[19,7616,7618],{"className":7617},[213],[19,7619,7621],{"className":7620,"style":2632},[217],[19,7622],{},[19,7624],{"className":7625,"style":2638},[558],[19,7627],{"className":7628,"style":2638},[558],[19,7630,7632],{"className":7631},[2588],[19,7633,7635,7669],{"className":7634},[208,209],[19,7636,7638,7666],{"className":7637},[213],[19,7639,7641,7652],{"className":7640,"style":2598},[217],[19,7642,7643,7646],{"style":2601},[19,7644],{"className":7645,"style":865},[225],[19,7647,7649],{"className":7648},[195],[19,7650,107],{"className":7651},[195],[19,7653,7654,7657],{"style":2613},[19,7655],{"className":7656,"style":865},[225],[19,7658,7660,7663],{"className":7659},[195],[19,7661,140],{"className":7662},[195],[19,7664,5238],{"className":7665},[195,199],[19,7667,241],{"className":7668},[240],[19,7670,7672],{"className":7671},[213],[19,7673,7675],{"className":7674,"style":2632},[217],[19,7676],{},[19,7678,7680],{"className":7679},[674],[19,7681,7683],{"className":7682},[281,282],[19,7684,7686,7706],{"className":7685},[208,209],[19,7687,7689,7703],{"className":7688},[213],[19,7690,7692],{"className":7691,"style":2598},[217],[19,7693,7694,7697],{"style":7533},[19,7695],{"className":7696,"style":7537},[225],[19,7698,7699],{"style":7540},[318,7700,7701],{"xmlns":320,"width":7543,"height":7544,"viewBox":7545},[327,7702],{"d":7548},[19,7704,241],{"className":7705},[240],[19,7707,7709],{"className":7708},[213],[19,7710,7712],{"className":7711,"style":2632},[217],[19,7713],{},[19,7715],{"className":7716,"style":255},[254],[19,7718,56],{"className":7719},[259],[19,7721],{"className":7722,"style":255},[254],[19,7724,7726,7730,7760,7763,7766],{"className":7725},[186],[19,7727],{"className":7728,"style":7729},[190],"height:0.9474em;vertical-align:-0.0833em;",[19,7731,7733,7736],{"className":7732},[195],[19,7734,5238],{"className":7735},[195,199],[19,7737,7739],{"className":7738},[204],[19,7740,7742],{"className":7741},[208],[19,7743,7745],{"className":7744},[213],[19,7746,7749],{"className":7747,"style":7748},[217],"height:0.8641em;",[19,7750,7751,7754],{"style":6430},[19,7752],{"className":7753,"style":226},[225],[19,7755,7757],{"className":7756},[230,231,232,233],[19,7758,158],{"className":7759},[195,233],[19,7761],{"className":7762,"style":487},[254],[19,7764,140],{"className":7765},[468],[19,7767],{"className":7768,"style":487},[254],[19,7770,7772,7776,7779,7782,7785],{"className":7771},[186],[19,7773],{"className":7774,"style":7775},[190],"height:0.7778em;vertical-align:-0.0833em;",[19,7777,5238],{"className":7778},[195,199],[19,7780],{"className":7781,"style":487},[254],[19,7783,140],{"className":7784},[468],[19,7786],{"className":7787,"style":487},[254],[19,7789,7791,7794,7797,7800,7803],{"className":7790},[186],[19,7792],{"className":7793,"style":1382},[190],[19,7795,107],{"className":7796},[195],[19,7798],{"className":7799,"style":255},[254],[19,7801,56],{"className":7802},[259],[19,7804],{"className":7805,"style":255},[254],[19,7807,7809,7812],{"className":7808},[186],[19,7810],{"className":7811,"style":1382},[190],[19,7813,78],{"className":7814},[195],[10,7816,7817,7818,5624,8045,5161],{},"Solving this equation, we obtain ",[19,7819,7821,7855],{"className":7820,"translate":23},[27],[19,7822,7824],{"className":7823},[31],[33,7825,7826],{"xmlns":35},[38,7827,7828,7852],{},[41,7829,7830,7836,7838],{},[44,7831,7832,7834],{},[47,7833,5238],{},[81,7835,107],{},[54,7837,56],{},[703,7839,7840,7850],{},[41,7841,7842,7844,7846],{},[81,7843,107],{},[54,7845,145],{},[708,7847,7848],{},[81,7849,712],{},[81,7851,158],{},[175,7853,7854],{"encoding":177},"\\lambda_1 = \\frac{1+\\sqrt{5}}{2}",[19,7856,7858,7913],{"className":7857,"ariaHidden":61},[182],[19,7859,7861,7864,7904,7907,7910],{"className":7860},[186],[19,7862],{"className":7863,"style":5582},[190],[19,7865,7867,7870],{"className":7866},[195],[19,7868,5238],{"className":7869},[195,199],[19,7871,7873],{"className":7872},[204],[19,7874,7876,7896],{"className":7875},[208,209],[19,7877,7879,7893],{"className":7878},[213],[19,7880,7882],{"className":7881,"style":450},[217],[19,7883,7884,7887],{"style":5603},[19,7885],{"className":7886,"style":226},[225],[19,7888,7890],{"className":7889},[230,231,232,233],[19,7891,107],{"className":7892},[195,233],[19,7894,241],{"className":7895},[240],[19,7897,7899],{"className":7898},[213],[19,7900,7902],{"className":7901,"style":248},[217],[19,7903],{},[19,7905],{"className":7906,"style":255},[254],[19,7908,56],{"className":7909},[259],[19,7911],{"className":7912,"style":255},[254],[19,7914,7916,7920],{"className":7915},[186],[19,7917],{"className":7918,"style":7919},[190],"height:1.383em;vertical-align:-0.345em;",[19,7921,7923,7926,8042],{"className":7922},[195],[19,7924],{"className":7925},[277,675],[19,7927,7929],{"className":7928},[703],[19,7930,7932,8033],{"className":7931},[208,209],[19,7933,7935,8030],{"className":7934},[213],[19,7936,7939,7954,7962],{"className":7937,"style":7938},[217],"height:1.038em;",[19,7940,7942,7945],{"style":7941},"top:-2.655em;",[19,7943],{"className":7944,"style":865},[225],[19,7946,7948],{"className":7947},[230,231,232,233],[19,7949,7951],{"className":7950},[195,233],[19,7952,158],{"className":7953},[195,233],[19,7955,7956,7959],{"style":932},[19,7957],{"className":7958,"style":865},[225],[19,7960],{"className":7961,"style":940},[939],[19,7963,7965,7968],{"style":7964},"top:-3.399em;",[19,7966],{"className":7967,"style":865},[225],[19,7969,7971],{"className":7970},[230,231,232,233],[19,7972,7974,7977,7980],{"className":7973},[195,233],[19,7975,107],{"className":7976},[195,233],[19,7978,145],{"className":7979},[468,233],[19,7981,7983],{"className":7982},[195,872,233],[19,7984,7986,8021],{"className":7985},[208,209],[19,7987,7989,8018],{"className":7988},[213],[19,7990,7993,8005],{"className":7991,"style":7992},[217],"height:0.9128em;",[19,7994,7996,7999],{"className":7995,"style":887},[886],[19,7997],{"className":7998,"style":865},[225],[19,8000,8002],{"className":8001,"style":894},[195,233],[19,8003,712],{"className":8004},[195,233],[19,8006,8008,8011],{"style":8007},"top:-2.8728em;",[19,8009],{"className":8010,"style":865},[225],[19,8012,8014],{"className":8013,"style":908},[907,233],[318,8015,8016],{"xmlns":320,"width":911,"height":912,"viewBox":913,"preserveAspectRatio":914},[327,8017],{"d":917},[19,8019,241],{"className":8020},[240],[19,8022,8024],{"className":8023},[213],[19,8025,8028],{"className":8026,"style":8027},[217],"height:0.1272em;",[19,8029],{},[19,8031,241],{"className":8032},[240],[19,8034,8036],{"className":8035},[213],[19,8037,8040],{"className":8038,"style":8039},[217],"height:0.345em;",[19,8041],{},[19,8043],{"className":8044},[674,675],[19,8046,8048,8082],{"className":8047,"translate":23},[27],[19,8049,8051],{"className":8050},[31],[33,8052,8053],{"xmlns":35},[38,8054,8055,8079],{},[41,8056,8057,8063,8065],{},[44,8058,8059,8061],{},[47,8060,5238],{},[81,8062,158],{},[54,8064,56],{},[703,8066,8067,8077],{},[41,8068,8069,8071,8073],{},[81,8070,107],{},[54,8072,140],{},[708,8074,8075],{},[81,8076,712],{},[81,8078,158],{},[175,8080,8081],{"encoding":177},"\\lambda_2 = \\frac{1-\\sqrt{5}}{2}",[19,8083,8085,8140],{"className":8084,"ariaHidden":61},[182],[19,8086,8088,8091,8131,8134,8137],{"className":8087},[186],[19,8089],{"className":8090,"style":5582},[190],[19,8092,8094,8097],{"className":8093},[195],[19,8095,5238],{"className":8096},[195,199],[19,8098,8100],{"className":8099},[204],[19,8101,8103,8123],{"className":8102},[208,209],[19,8104,8106,8120],{"className":8105},[213],[19,8107,8109],{"className":8108,"style":450},[217],[19,8110,8111,8114],{"style":5603},[19,8112],{"className":8113,"style":226},[225],[19,8115,8117],{"className":8116},[230,231,232,233],[19,8118,158],{"className":8119},[195,233],[19,8121,241],{"className":8122},[240],[19,8124,8126],{"className":8125},[213],[19,8127,8129],{"className":8128,"style":248},[217],[19,8130],{},[19,8132],{"className":8133,"style":255},[254],[19,8135,56],{"className":8136},[259],[19,8138],{"className":8139,"style":255},[254],[19,8141,8143,8146],{"className":8142},[186],[19,8144],{"className":8145,"style":7919},[190],[19,8147,8149,8152,8261],{"className":8148},[195],[19,8150],{"className":8151},[277,675],[19,8153,8155],{"className":8154},[703],[19,8156,8158,8253],{"className":8157},[208,209],[19,8159,8161,8250],{"className":8160},[213],[19,8162,8164,8178,8186],{"className":8163,"style":7938},[217],[19,8165,8166,8169],{"style":7941},[19,8167],{"className":8168,"style":865},[225],[19,8170,8172],{"className":8171},[230,231,232,233],[19,8173,8175],{"className":8174},[195,233],[19,8176,158],{"className":8177},[195,233],[19,8179,8180,8183],{"style":932},[19,8181],{"className":8182,"style":865},[225],[19,8184],{"className":8185,"style":940},[939],[19,8187,8188,8191],{"style":7964},[19,8189],{"className":8190,"style":865},[225],[19,8192,8194],{"className":8193},[230,231,232,233],[19,8195,8197,8200,8203],{"className":8196},[195,233],[19,8198,107],{"className":8199},[195,233],[19,8201,140],{"className":8202},[468,233],[19,8204,8206],{"className":8205},[195,872,233],[19,8207,8209,8242],{"className":8208},[208,209],[19,8210,8212,8239],{"className":8211},[213],[19,8213,8215,8227],{"className":8214,"style":7992},[217],[19,8216,8218,8221],{"className":8217,"style":887},[886],[19,8219],{"className":8220,"style":865},[225],[19,8222,8224],{"className":8223,"style":894},[195,233],[19,8225,712],{"className":8226},[195,233],[19,8228,8229,8232],{"style":8007},[19,8230],{"className":8231,"style":865},[225],[19,8233,8235],{"className":8234,"style":908},[907,233],[318,8236,8237],{"xmlns":320,"width":911,"height":912,"viewBox":913,"preserveAspectRatio":914},[327,8238],{"d":917},[19,8240,241],{"className":8241},[240],[19,8243,8245],{"className":8244},[213],[19,8246,8248],{"className":8247,"style":8027},[217],[19,8249],{},[19,8251,241],{"className":8252},[240],[19,8254,8256],{"className":8255},[213],[19,8257,8259],{"className":8258,"style":8039},[217],[19,8260],{},[19,8262],{"className":8263},[674,675],[10,8265,8266],{},"Therefore, the diagonal matrix is:",[19,8268,8270],{"className":8269,"translate":23},[22],[19,8271,8273,8375],{"className":8272,"translate":23},[27],[19,8274,8276],{"className":8275},[31],[33,8277,8278],{"xmlns":35,"display":36},[38,8279,8280,8372],{},[41,8281,8282,8292,8294,8300,8302,8308,8310,8312],{},[41,8283,8284,8286,8288,8290],{},[47,8285,5771],{"mathvariant":5770},[47,8287,5774],{"mathvariant":5770},[47,8289,5777],{"mathvariant":5770},[47,8291,5780],{"mathvariant":5770},[54,8293,62],{"stretchy":79},[44,8295,8296,8298],{},[47,8297,5238],{},[81,8299,107],{},[54,8301,5791],{"separator":61},[44,8303,8304,8306],{},[47,8305,5238],{},[81,8307,158],{},[54,8309,5800],{"stretchy":79},[54,8311,56],{},[41,8313,8314,8316,8370],{},[54,8315,2436],{"fence":61},[64,8317,8318,8344],{"rowspacing":2439,"columnalign":2440,"columnspacing":68},[70,8319,8320,8338],{},[73,8321,8322],{},[76,8323,8324],{"scriptlevel":78,"displaystyle":79},[703,8325,8326,8336],{},[41,8327,8328,8330,8332],{},[81,8329,107],{},[54,8331,145],{},[708,8333,8334],{},[81,8335,712],{},[81,8337,158],{},[73,8339,8340],{},[76,8341,8342],{"scriptlevel":78,"displaystyle":79},[81,8343,78],{},[70,8345,8346,8352],{},[73,8347,8348],{},[76,8349,8350],{"scriptlevel":78,"displaystyle":79},[81,8351,78],{},[73,8353,8354],{},[76,8355,8356],{"scriptlevel":78,"displaystyle":79},[703,8357,8358,8368],{},[41,8359,8360,8362,8364],{},[81,8361,107],{},[54,8363,140],{},[708,8365,8366],{},[81,8367,712],{},[81,8369,158],{},[54,8371,2471],{"fence":61},[175,8373,8374],{"encoding":177},"\\rm{diag}\\{\\lambda_1, \\lambda_2\\} = \\begin{bmatrix}\n\\frac{1+\\sqrt{5}}{2} & 0 \\\\\n0 & \\frac{1-\\sqrt{5}}{2}\n\\end{bmatrix}",[19,8376,8378],{"className":8377,"ariaHidden":61},[182],[19,8379,8381,8384],{"className":8380},[186],[19,8382],{"className":8383,"style":2016},[190],[19,8385,8387,8393,8396,8436,8439,8442,8482,8485,8488,8491,8494],{"className":8386},[195],[19,8388,8390],{"className":8389},[195],[19,8391,5824],{"className":8392,"style":5823},[195,5822],[19,8394,62],{"className":8395},[277],[19,8397,8399,8402],{"className":8398},[195],[19,8400,5238],{"className":8401},[195,199],[19,8403,8405],{"className":8404},[204],[19,8406,8408,8428],{"className":8407},[208,209],[19,8409,8411,8425],{"className":8410},[213],[19,8412,8414],{"className":8413,"style":450},[217],[19,8415,8416,8419],{"style":5603},[19,8417],{"className":8418,"style":226},[225],[19,8420,8422],{"className":8421},[230,231,232,233],[19,8423,107],{"className":8424},[195,5822,233],[19,8426,241],{"className":8427},[240],[19,8429,8431],{"className":8430},[213],[19,8432,8434],{"className":8433,"style":248},[217],[19,8435],{},[19,8437,5791],{"className":8438},[5871],[19,8440],{"className":8441,"style":971},[254],[19,8443,8445,8448],{"className":8444},[195],[19,8446,5238],{"className":8447},[195,199],[19,8449,8451],{"className":8450},[204],[19,8452,8454,8474],{"className":8453},[208,209],[19,8455,8457,8471],{"className":8456},[213],[19,8458,8460],{"className":8459,"style":450},[217],[19,8461,8462,8465],{"style":5603},[19,8463],{"className":8464,"style":226},[225],[19,8466,8468],{"className":8467},[230,231,232,233],[19,8469,158],{"className":8470},[195,5822,233],[19,8472,241],{"className":8473},[240],[19,8475,8477],{"className":8476},[213],[19,8478,8480],{"className":8479,"style":248},[217],[19,8481],{},[19,8483,5800],{"className":8484},[674],[19,8486],{"className":8487,"style":255},[254],[19,8489,56],{"className":8490},[259],[19,8492],{"className":8493,"style":255},[254],[19,8495,8497,8503,8839],{"className":8496},[273],[19,8498,8500],{"className":8499,"style":979},[277,978],[19,8501,2436],{"className":8502},[281,983],[19,8504,8506],{"className":8505},[195],[19,8507,8509,8673,8676,8679],{"className":8508},[64],[19,8510,8512],{"className":8511},[2588],[19,8513,8515,8664],{"className":8514},[208,209],[19,8516,8518,8661],{"className":8517},[213],[19,8519,8522,8650],{"className":8520,"style":8521},[217],"height:1.648em;",[19,8523,8525,8529],{"style":8524},"top:-3.648em;",[19,8526],{"className":8527,"style":8528},[225],"height:3.038em;",[19,8530,8532],{"className":8531},[195],[19,8533,8535,8538,8647],{"className":8534},[195],[19,8536],{"className":8537},[277,675],[19,8539,8541],{"className":8540},[703],[19,8542,8544,8639],{"className":8543},[208,209],[19,8545,8547,8636],{"className":8546},[213],[19,8548,8550,8564,8572],{"className":8549,"style":7938},[217],[19,8551,8552,8555],{"style":7941},[19,8553],{"className":8554,"style":865},[225],[19,8556,8558],{"className":8557},[230,231,232,233],[19,8559,8561],{"className":8560},[195,233],[19,8562,158],{"className":8563},[195,233],[19,8565,8566,8569],{"style":932},[19,8567],{"className":8568,"style":865},[225],[19,8570],{"className":8571,"style":940},[939],[19,8573,8574,8577],{"style":7964},[19,8575],{"className":8576,"style":865},[225],[19,8578,8580],{"className":8579},[230,231,232,233],[19,8581,8583,8586,8589],{"className":8582},[195,233],[19,8584,107],{"className":8585},[195,233],[19,8587,145],{"className":8588},[468,233],[19,8590,8592],{"className":8591},[195,872,233],[19,8593,8595,8628],{"className":8594},[208,209],[19,8596,8598,8625],{"className":8597},[213],[19,8599,8601,8613],{"className":8600,"style":7992},[217],[19,8602,8604,8607],{"className":8603,"style":887},[886],[19,8605],{"className":8606,"style":865},[225],[19,8608,8610],{"className":8609,"style":894},[195,233],[19,8611,712],{"className":8612},[195,233],[19,8614,8615,8618],{"style":8007},[19,8616],{"className":8617,"style":865},[225],[19,8619,8621],{"className":8620,"style":908},[907,233],[318,8622,8623],{"xmlns":320,"width":911,"height":912,"viewBox":913,"preserveAspectRatio":914},[327,8624],{"d":917},[19,8626,241],{"className":8627},[240],[19,8629,8631],{"className":8630},[213],[19,8632,8634],{"className":8633,"style":8027},[217],[19,8635],{},[19,8637,241],{"className":8638},[240],[19,8640,8642],{"className":8641},[213],[19,8643,8645],{"className":8644,"style":8039},[217],[19,8646],{},[19,8648],{"className":8649},[674,675],[19,8651,8652,8655],{"style":2215},[19,8653],{"className":8654,"style":8528},[225],[19,8656,8658],{"className":8657},[195],[19,8659,78],{"className":8660},[195],[19,8662,241],{"className":8663},[240],[19,8665,8667],{"className":8666},[213],[19,8668,8671],{"className":8669,"style":8670},[217],"height:1.148em;",[19,8672],{},[19,8674],{"className":8675,"style":2638},[558],[19,8677],{"className":8678,"style":2638},[558],[19,8680,8682],{"className":8681},[2588],[19,8683,8685,8831],{"className":8684},[208,209],[19,8686,8688,8828],{"className":8687},[213],[19,8689,8691,8702],{"className":8690,"style":8521},[217],[19,8692,8693,8696],{"style":8524},[19,8694],{"className":8695,"style":8528},[225],[19,8697,8699],{"className":8698},[195],[19,8700,78],{"className":8701},[195],[19,8703,8704,8707],{"style":2215},[19,8705],{"className":8706,"style":8528},[225],[19,8708,8710],{"className":8709},[195],[19,8711,8713,8716,8825],{"className":8712},[195],[19,8714],{"className":8715},[277,675],[19,8717,8719],{"className":8718},[703],[19,8720,8722,8817],{"className":8721},[208,209],[19,8723,8725,8814],{"className":8724},[213],[19,8726,8728,8742,8750],{"className":8727,"style":7938},[217],[19,8729,8730,8733],{"style":7941},[19,8731],{"className":8732,"style":865},[225],[19,8734,8736],{"className":8735},[230,231,232,233],[19,8737,8739],{"className":8738},[195,233],[19,8740,158],{"className":8741},[195,233],[19,8743,8744,8747],{"style":932},[19,8745],{"className":8746,"style":865},[225],[19,8748],{"className":8749,"style":940},[939],[19,8751,8752,8755],{"style":7964},[19,8753],{"className":8754,"style":865},[225],[19,8756,8758],{"className":8757},[230,231,232,233],[19,8759,8761,8764,8767],{"className":8760},[195,233],[19,8762,107],{"className":8763},[195,233],[19,8765,140],{"className":8766},[468,233],[19,8768,8770],{"className":8769},[195,872,233],[19,8771,8773,8806],{"className":8772},[208,209],[19,8774,8776,8803],{"className":8775},[213],[19,8777,8779,8791],{"className":8778,"style":7992},[217],[19,8780,8782,8785],{"className":8781,"style":887},[886],[19,8783],{"className":8784,"style":865},[225],[19,8786,8788],{"className":8787,"style":894},[195,233],[19,8789,712],{"className":8790},[195,233],[19,8792,8793,8796],{"style":8007},[19,8794],{"className":8795,"style":865},[225],[19,8797,8799],{"className":8798,"style":908},[907,233],[318,8800,8801],{"xmlns":320,"width":911,"height":912,"viewBox":913,"preserveAspectRatio":914},[327,8802],{"d":917},[19,8804,241],{"className":8805},[240],[19,8807,8809],{"className":8808},[213],[19,8810,8812],{"className":8811,"style":8027},[217],[19,8813],{},[19,8815,241],{"className":8816},[240],[19,8818,8820],{"className":8819},[213],[19,8821,8823],{"className":8822,"style":8039},[217],[19,8824],{},[19,8826],{"className":8827},[674,675],[19,8829,241],{"className":8830},[240],[19,8832,8834],{"className":8833},[213],[19,8835,8837],{"className":8836,"style":8670},[217],[19,8838],{},[19,8840,8842],{"className":8841,"style":979},[674,978],[19,8843,2471],{"className":8844},[281,983],[10,8846,8847,8848,9058,9059,5624,9128,9197,9198,9236],{},"Assuming the eigenvector is ",[19,8849,8851,8895],{"className":8850,"translate":23},[27],[19,8852,8854],{"className":8853},[31],[33,8855,8856],{"xmlns":35},[38,8857,8858,8892],{},[41,8859,8860,8862,8864],{},[47,8861,5233],{"mathvariant":4730},[54,8863,56],{},[719,8865,8866,8889],{},[41,8867,8868,8870,8887],{},[54,8869,2436],{"fence":61},[64,8871,8872],{"rowspacing":2439,"columnalign":2440,"columnspacing":68},[70,8873,8874,8880],{},[73,8875,8876],{},[76,8877,8878],{"scriptlevel":78,"displaystyle":79},[47,8879,5233],{},[73,8881,8882],{},[76,8883,8884],{"scriptlevel":78,"displaystyle":79},[47,8885,8886],{},"y",[54,8888,2471],{"fence":61},[47,8890,8891],{},"T",[175,8893,8894],{"encoding":177},"\\boldsymbol{x} = \\begin{bmatrix}\nx & y\n\\end{bmatrix}^T",[19,8896,8898,8923],{"className":8897,"ariaHidden":61},[182],[19,8899,8901,8905,8914,8917,8920],{"className":8900},[186],[19,8902],{"className":8903,"style":8904},[190],"height:0.4444em;",[19,8906,8908],{"className":8907},[195],[19,8909,8911],{"className":8910},[195],[19,8912,5233],{"className":8913},[195,4792],[19,8915],{"className":8916,"style":255},[254],[19,8918,56],{"className":8919},[259],[19,8921],{"className":8922,"style":255},[254],[19,8924,8926,8930],{"className":8925},[186],[19,8927],{"className":8928,"style":8929},[190],"height:1.4312em;vertical-align:-0.35em;",[19,8931,8933,9033],{"className":8932},[273],[19,8934,8936,8943,9027],{"className":8935},[273],[19,8937,8939],{"className":8938,"style":979},[277,978],[19,8940,2436],{"className":8941},[281,8942],"size1",[19,8944,8946],{"className":8945},[195],[19,8947,8949,8986,8989,8992],{"className":8948},[64],[19,8950,8952],{"className":8951},[2588],[19,8953,8955,8977],{"className":8954},[208,209],[19,8956,8958,8974],{"className":8957},[213],[19,8959,8962],{"className":8960,"style":8961},[217],"height:0.85em;",[19,8963,8965,8968],{"style":8964},"top:-3.01em;",[19,8966],{"className":8967,"style":865},[225],[19,8969,8971],{"className":8970},[195],[19,8972,5233],{"className":8973},[195,199],[19,8975,241],{"className":8976},[240],[19,8978,8980],{"className":8979},[213],[19,8981,8984],{"className":8982,"style":8983},[217],"height:0.35em;",[19,8985],{},[19,8987],{"className":8988,"style":2638},[558],[19,8990],{"className":8991,"style":2638},[558],[19,8993,8995],{"className":8994},[2588],[19,8996,8998,9019],{"className":8997},[208,209],[19,8999,9001,9016],{"className":9000},[213],[19,9002,9004],{"className":9003,"style":8961},[217],[19,9005,9006,9009],{"style":8964},[19,9007],{"className":9008,"style":865},[225],[19,9010,9012],{"className":9011},[195],[19,9013,8886],{"className":9014,"style":9015},[195,199],"margin-right:0.03588em;",[19,9017,241],{"className":9018},[240],[19,9020,9022],{"className":9021},[213],[19,9023,9025],{"className":9024,"style":8983},[217],[19,9026],{},[19,9028,9030],{"className":9029,"style":979},[674,978],[19,9031,2471],{"className":9032},[281,8942],[19,9034,9036],{"className":9035},[204],[19,9037,9039],{"className":9038},[208],[19,9040,9042],{"className":9041},[213],[19,9043,9046],{"className":9044,"style":9045},[217],"height:1.0812em;",[19,9047,9049,9052],{"style":9048},"top:-3.3029em;margin-right:0.05em;",[19,9050],{"className":9051,"style":226},[225],[19,9053,9055],{"className":9054},[230,231,232,233],[19,9056,8891],{"className":9057,"style":200},[195,199,233],", we can substitute ",[19,9060,9062,9079],{"className":9061,"translate":23},[27],[19,9063,9065],{"className":9064},[31],[33,9066,9067],{"xmlns":35},[38,9068,9069,9077],{},[41,9070,9071],{},[44,9072,9073,9075],{},[47,9074,5238],{},[81,9076,107],{},[175,9078,5572],{"encoding":177},[19,9080,9082],{"className":9081,"ariaHidden":61},[182],[19,9083,9085,9088],{"className":9084},[186],[19,9086],{"className":9087,"style":5582},[190],[19,9089,9091,9094],{"className":9090},[195],[19,9092,5238],{"className":9093},[195,199],[19,9095,9097],{"className":9096},[204],[19,9098,9100,9120],{"className":9099},[208,209],[19,9101,9103,9117],{"className":9102},[213],[19,9104,9106],{"className":9105,"style":450},[217],[19,9107,9108,9111],{"style":5603},[19,9109],{"className":9110,"style":226},[225],[19,9112,9114],{"className":9113},[230,231,232,233],[19,9115,107],{"className":9116},[195,233],[19,9118,241],{"className":9119},[240],[19,9121,9123],{"className":9122},[213],[19,9124,9126],{"className":9125,"style":248},[217],[19,9127],{},[19,9129,9131,9148],{"className":9130,"translate":23},[27],[19,9132,9134],{"className":9133},[31],[33,9135,9136],{"xmlns":35},[38,9137,9138,9146],{},[41,9139,9140],{},[44,9141,9142,9144],{},[47,9143,5238],{},[81,9145,158],{},[175,9147,5645],{"encoding":177},[19,9149,9151],{"className":9150,"ariaHidden":61},[182],[19,9152,9154,9157],{"className":9153},[186],[19,9155],{"className":9156,"style":5582},[190],[19,9158,9160,9163],{"className":9159},[195],[19,9161,5238],{"className":9162},[195,199],[19,9164,9166],{"className":9165},[204],[19,9167,9169,9189],{"className":9168},[208,209],[19,9170,9172,9186],{"className":9171},[213],[19,9173,9175],{"className":9174,"style":450},[217],[19,9176,9177,9180],{"style":5603},[19,9178],{"className":9179,"style":226},[225],[19,9181,9183],{"className":9182},[230,231,232,233],[19,9184,158],{"className":9185},[195,233],[19,9187,241],{"className":9188},[240],[19,9190,9192],{"className":9191},[213],[19,9193,9195],{"className":9194,"style":248},[217],[19,9196],{}," into equation ",[19,9199,9201,9218],{"className":9200,"translate":23},[27],[19,9202,9204],{"className":9203},[31],[33,9205,9206],{"xmlns":35},[38,9207,9208,9216],{},[41,9209,9210,9212,9214],{},[54,9211,717],{"stretchy":79},[81,9213,158],{},[54,9215,741],{"stretchy":79},[175,9217,6333],{"encoding":177},[19,9219,9221],{"className":9220,"ariaHidden":61},[182],[19,9222,9224,9227,9230,9233],{"className":9223},[186],[19,9225],{"className":9226,"style":5312},[190],[19,9228,717],{"className":9229},[277],[19,9231,158],{"className":9232},[195],[19,9234,741],{"className":9235},[674]," to obtain two systems of equations:",[19,9238,9240],{"className":9239,"translate":23},[22],[19,9241,9243,9381],{"className":9242,"translate":23},[27],[19,9244,9246],{"className":9245},[31],[33,9247,9248],{"xmlns":35,"display":36},[38,9249,9250,9378],{},[41,9251,9252,9314,9316],{},[41,9253,9254,9256],{},[54,9255,62],{"fence":61},[64,9257,9258,9290],{"rowspacing":66,"columnalign":67,"columnspacing":68},[70,9259,9260],{},[73,9261,9262],{},[76,9263,9264],{"scriptlevel":78,"displaystyle":79},[41,9265,9266,9268,9270,9272,9278,9280,9282,9284,9286,9288],{},[54,9267,717],{"stretchy":79},[81,9269,107],{},[54,9271,140],{},[44,9273,9274,9276],{},[47,9275,5238],{},[81,9277,107],{},[54,9279,741],{"stretchy":79},[47,9281,5233],{},[54,9283,145],{},[47,9285,8886],{},[54,9287,56],{},[81,9289,78],{},[70,9291,9292],{},[73,9293,9294],{},[76,9295,9296],{"scriptlevel":78,"displaystyle":79},[41,9297,9298,9300,9302,9308,9310,9312],{},[47,9299,5233],{},[54,9301,140],{},[44,9303,9304,9306],{},[47,9305,5238],{},[81,9307,107],{},[47,9309,8886],{},[54,9311,56],{},[81,9313,78],{},[54,9315,5791],{"separator":61},[41,9317,9318,9320],{},[54,9319,62],{"fence":61},[64,9321,9322,9354],{"rowspacing":66,"columnalign":67,"columnspacing":68},[70,9323,9324],{},[73,9325,9326],{},[76,9327,9328],{"scriptlevel":78,"displaystyle":79},[41,9329,9330,9332,9334,9336,9342,9344,9346,9348,9350,9352],{},[54,9331,717],{"stretchy":79},[81,9333,107],{},[54,9335,140],{},[44,9337,9338,9340],{},[47,9339,5238],{},[81,9341,158],{},[54,9343,741],{"stretchy":79},[47,9345,5233],{},[54,9347,145],{},[47,9349,8886],{},[54,9351,56],{},[81,9353,78],{},[70,9355,9356],{},[73,9357,9358],{},[76,9359,9360],{"scriptlevel":78,"displaystyle":79},[41,9361,9362,9364,9366,9372,9374,9376],{},[47,9363,5233],{},[54,9365,140],{},[44,9367,9368,9370],{},[47,9369,5238],{},[81,9371,158],{},[47,9373,8886],{},[54,9375,56],{},[81,9377,78],{},[175,9379,9380],{"encoding":177},"\\begin{cases}\n(1-\\lambda_1)x + y = 0 \\\\\nx - \\lambda_1y = 0\n\\end{cases},\n\\begin{cases}\n(1-\\lambda_2)x + y = 0 \\\\\nx - \\lambda_2y = 0\n\\end{cases}",[19,9382,9384],{"className":9383,"ariaHidden":61},[182],[19,9385,9387,9390,9599,9602,9605,9608],{"className":9386},[186],[19,9388],{"className":9389,"style":2016},[190],[19,9391,9393,9399,9596],{"className":9392},[273],[19,9394,9396],{"className":9395,"style":979},[277,978],[19,9397,62],{"className":9398},[281,983],[19,9400,9402],{"className":9401},[195],[19,9403,9405],{"className":9404},[64],[19,9406,9408],{"className":9407},[387],[19,9409,9411,9588],{"className":9410},[208,209],[19,9412,9414,9585],{"className":9413},[213],[19,9415,9417,9510],{"className":9416,"style":2044},[217],[19,9418,9419,9422],{"style":2047},[19,9420],{"className":9421,"style":404},[225],[19,9423,9425,9428,9431,9434,9437,9440,9480,9483,9486,9489,9492,9495,9498,9501,9504,9507],{"className":9424},[195],[19,9426,717],{"className":9427},[277],[19,9429,107],{"className":9430},[195],[19,9432],{"className":9433,"style":487},[254],[19,9435,140],{"className":9436},[468],[19,9438],{"className":9439,"style":487},[254],[19,9441,9443,9446],{"className":9442},[195],[19,9444,5238],{"className":9445},[195,199],[19,9447,9449],{"className":9448},[204],[19,9450,9452,9472],{"className":9451},[208,209],[19,9453,9455,9469],{"className":9454},[213],[19,9456,9458],{"className":9457,"style":450},[217],[19,9459,9460,9463],{"style":5603},[19,9461],{"className":9462,"style":226},[225],[19,9464,9466],{"className":9465},[230,231,232,233],[19,9467,107],{"className":9468},[195,233],[19,9470,241],{"className":9471},[240],[19,9473,9475],{"className":9474},[213],[19,9476,9478],{"className":9477,"style":248},[217],[19,9479],{},[19,9481,741],{"className":9482},[674],[19,9484,5233],{"className":9485},[195,199],[19,9487],{"className":9488,"style":487},[254],[19,9490,145],{"className":9491},[468],[19,9493],{"className":9494,"style":487},[254],[19,9496,8886],{"className":9497,"style":9015},[195,199],[19,9499],{"className":9500,"style":255},[254],[19,9502,56],{"className":9503},[259],[19,9505],{"className":9506,"style":255},[254],[19,9508,78],{"className":9509},[195],[19,9511,9512,9515],{"style":2215},[19,9513],{"className":9514,"style":404},[225],[19,9516,9518,9521,9524,9527,9530,9570,9573,9576,9579,9582],{"className":9517},[195],[19,9519,5233],{"className":9520},[195,199],[19,9522],{"className":9523,"style":487},[254],[19,9525,140],{"className":9526},[468],[19,9528],{"className":9529,"style":487},[254],[19,9531,9533,9536],{"className":9532},[195],[19,9534,5238],{"className":9535},[195,199],[19,9537,9539],{"className":9538},[204],[19,9540,9542,9562],{"className":9541},[208,209],[19,9543,9545,9559],{"className":9544},[213],[19,9546,9548],{"className":9547,"style":450},[217],[19,9549,9550,9553],{"style":5603},[19,9551],{"className":9552,"style":226},[225],[19,9554,9556],{"className":9555},[230,231,232,233],[19,9557,107],{"className":9558},[195,233],[19,9560,241],{"className":9561},[240],[19,9563,9565],{"className":9564},[213],[19,9566,9568],{"className":9567,"style":248},[217],[19,9569],{},[19,9571,8886],{"className":9572,"style":9015},[195,199],[19,9574],{"className":9575,"style":255},[254],[19,9577,56],{"className":9578},[259],[19,9580],{"className":9581,"style":255},[254],[19,9583,78],{"className":9584},[195],[19,9586,241],{"className":9587},[240],[19,9589,9591],{"className":9590},[213],[19,9592,9594],{"className":9593,"style":2408},[217],[19,9595],{},[19,9597],{"className":9598},[674,675],[19,9600],{"className":9601,"style":971},[254],[19,9603,5791],{"className":9604},[5871],[19,9606],{"className":9607,"style":971},[254],[19,9609,9611,9617,9814],{"className":9610},[273],[19,9612,9614],{"className":9613,"style":979},[277,978],[19,9615,62],{"className":9616},[281,983],[19,9618,9620],{"className":9619},[195],[19,9621,9623],{"className":9622},[64],[19,9624,9626],{"className":9625},[387],[19,9627,9629,9806],{"className":9628},[208,209],[19,9630,9632,9803],{"className":9631},[213],[19,9633,9635,9728],{"className":9634,"style":2044},[217],[19,9636,9637,9640],{"style":2047},[19,9638],{"className":9639,"style":404},[225],[19,9641,9643,9646,9649,9652,9655,9658,9698,9701,9704,9707,9710,9713,9716,9719,9722,9725],{"className":9642},[195],[19,9644,717],{"className":9645},[277],[19,9647,107],{"className":9648},[195],[19,9650],{"className":9651,"style":487},[254],[19,9653,140],{"className":9654},[468],[19,9656],{"className":9657,"style":487},[254],[19,9659,9661,9664],{"className":9660},[195],[19,9662,5238],{"className":9663},[195,199],[19,9665,9667],{"className":9666},[204],[19,9668,9670,9690],{"className":9669},[208,209],[19,9671,9673,9687],{"className":9672},[213],[19,9674,9676],{"className":9675,"style":450},[217],[19,9677,9678,9681],{"style":5603},[19,9679],{"className":9680,"style":226},[225],[19,9682,9684],{"className":9683},[230,231,232,233],[19,9685,158],{"className":9686},[195,233],[19,9688,241],{"className":9689},[240],[19,9691,9693],{"className":9692},[213],[19,9694,9696],{"className":9695,"style":248},[217],[19,9697],{},[19,9699,741],{"className":9700},[674],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",[19,9821,9823,9841],{"className":9822,"translate":23},[27],[19,9824,9826],{"className":9825},[31],[33,9827,9828],{"xmlns":35},[38,9829,9830,9838],{},[41,9831,9832,9834,9836],{},[47,9833,8886],{},[54,9835,56],{},[81,9837,107],{},[175,9839,9840],{"encoding":177},"y = 1",[19,9842,9844,9863],{"className":9843,"ariaHidden":61},[182],[19,9845,9847,9851,9854,9857,9860],{"className":9846},[186],[19,9848],{"className":9849,"style":9850},[190],"height:0.625em;vertical-align:-0.1944em;",[19,9852,8886],{"className":9853,"style":9015},[195,199],[19,9855],{"className":9856,"style":255},[254],[19,9858,56],{"className":9859},[259],[19,9861],{"className":9862,"style":255},[254],[19,9864,9866,9869],{"className":9865},[186],[19,9867],{"className":9868,"style":1382},[190],[19,9870,107],{"className":9871},[195]," in both systems of equations gives us the two eigenvectors of the matrix:",[19,9874,9876],{"className":9875,"translate":23},[22],[19,9877,9879,9985],{"className":9878,"translate":23},[27],[19,9880,9882],{"className":9881},[31],[33,9883,9884],{"xmlns":35,"display":36},[38,9885,9886,9982],{},[41,9887,9888,9894,9896,9934,9936,9942,9944],{},[44,9889,9890,9892],{},[47,9891,5233],{"mathvariant":4730},[81,9893,107],{},[54,9895,56],{},[719,9897,9898,9932],{},[41,9899,9900,9902,9930],{},[54,9901,2436],{"fence":61},[64,9903,9904],{"rowspacing":2439,"columnalign":2440,"columnspacing":68},[70,9905,9906,9924],{},[73,9907,9908],{},[76,9909,9910],{"scriptlevel":78,"displaystyle":79},[703,9911,9912,9922],{},[41,9913,9914,9916,9918],{},[81,9915,107],{},[54,9917,145],{},[708,9919,9920],{},[81,9921,712],{},[81,9923,158],{},[73,9925,9926],{},[76,9927,9928],{"scriptlevel":78,"displaystyle":79},[81,9929,107],{},[54,9931,2471],{"fence":61},[47,9933,8891],{},[54,9935,5791],{"separator":61},[44,9937,9938,9940],{},[47,9939,5233],{"mathvariant":4730},[81,9941,158],{},[54,9943,56],{},[719,9945,9946,9980],{},[41,9947,9948,9950,9978],{},[54,9949,2436],{"fence":61},[64,9951,9952],{"rowspacing":2439,"columnalign":2440,"columnspacing":68},[70,9953,9954,9972],{},[73,9955,9956],{},[76,9957,9958],{"scriptlevel":78,"displaystyle":79},[703,9959,9960,9970],{},[41,9961,9962,9964,9966],{},[81,9963,107],{},[54,9965,140],{},[708,9967,9968],{},[81,9969,712],{},[81,9971,158],{},[73,9973,9974],{},[76,9975,9976],{"scriptlevel":78,"displaystyle":79},[81,9977,107],{},[54,9979,2471],{"fence":61},[47,9981,8891],{},[175,9983,9984],{"encoding":177},"\\boldsymbol{x}_1 = \\begin{bmatrix}\n\\frac{1+\\sqrt{5}}{2} & 1\n\\end{bmatrix}^T,\n\\boldsymbol{x}_2 = \\begin{bmatrix}\n\\frac{1-\\sqrt{5}}{2} & 1\n\\end{bmatrix}^T",[19,9986,9988,10051,10364],{"className":9987,"ariaHidden":61},[182],[19,9989,9991,9995,10042,10045,10048],{"className":9990},[186],[19,9992],{"className":9993,"style":9994},[190],"height:0.5944em;vertical-align:-0.15em;",[19,9996,9998,10007],{"className":9997},[195],[19,9999,10001],{"className":10000},[195],[19,10002,10004],{"className":10003},[195],[19,10005,5233],{"className":10006},[195,4792],[19,10008,10010],{"className":10009},[204],[19,10011,10013,10034],{"className":10012},[208,209],[19,10014,10016,10031],{"className":10015},[213],[19,10017,10019],{"className":10018,"style":450},[217],[19,10020,10022,10025],{"style":10021},"top:-2.55em;margin-right:0.05em;",[19,10023],{"className":10024,"style":226},[225],[19,10026,10028],{"className":10027},[230,231,232,233],[19,10029,107],{"className":10030},[195,233],[19,10032,241],{"className":10033},[240],[19,10035,10037],{"className":10036},[213],[19,10038,10040],{"className":10039,"style":248},[217],[19,10041],{},[19,10043],{"className":10044,"style":255},[254],[19,10046,56],{"className":10047},[259],[19,10049],{"className":10050,"style":255},[254],[19,10052,10054,10058,10300,10303,10306,10309,10355,10358,10361],{"className":10053},[186],[19,10055],{"className":10056,"style":10057},[190],"height:2.0313em;vertical-align:-0.65em;",[19,10059,10061,10275],{"className":10060},[273],[19,10062,10064,10071,10269],{"className":10063},[273],[19,10065,10067],{"className":10066,"style":979},[277,978],[19,10068,2436],{"className":10069},[281,10070],"size2",[19,10072,10074],{"className":10073},[195],[19,10075,10077,10229,10232,10235],{"className":10076},[64],[19,10078,10080],{"className":10079},[2588],[19,10081,10083,10220],{"className":10082},[208,209],[19,10084,10086,10217],{"className":10085},[213],[19,10087,10090],{"className":10088,"style":10089},[217],"height:0.949em;",[19,10091,10093,10096],{"style":10092},"top:-2.949em;",[19,10094],{"className":10095,"style":8528},[225],[19,10097,10099],{"className":10098},[195],[19,10100,10102,10105,10214],{"className":10101},[195],[19,10103],{"className":10104},[277,675],[19,10106,10108],{"className":10107},[703],[19,10109,10111,10206],{"className":10110},[208,209],[19,10112,10114,10203],{"className":10113},[213],[19,10115,10117,10131,10139],{"className":10116,"style":7938},[217],[19,10118,10119,10122],{"style":7941},[19,10120],{"className":10121,"style":865},[225],[19,10123,10125],{"className":10124},[230,231,232,233],[19,10126,10128],{"className":10127},[195,233],[19,10129,158],{"className":10130},[195,233],[19,10132,10133,10136],{"style":932},[19,10134],{"className":10135,"style":865},[225],[19,10137],{"className":10138,"style":940},[939],[19,10140,10141,10144],{"style":7964},[19,10142],{"className":10143,"style":865},[225],[19,10145,10147],{"className":10146},[230,231,232,233],[19,10148,10150,10153,10156],{"className":10149},[195,233],[19,10151,107],{"className":10152},[195,233],[19,10154,145],{"className":10155},[468,233],[19,10157,10159],{"className":10158},[195,872,233],[19,10160,10162,10195],{"className":10161},[208,209],[19,10163,10165,10192],{"className":10164},[213],[19,10166,10168,10180],{"className":10167,"style":7992},[217],[19,10169,10171,10174],{"className":10170,"style":887},[886],[19,10172],{"className":10173,"style":865},[225],[19,10175,10177],{"className":10176,"style":894},[195,233],[19,10178,712],{"className":10179},[195,233],[19,10181,10182,10185],{"style":8007},[19,10183],{"className":10184,"style":865},[225],[19,10186,10188],{"className":10187,"style":908},[907,233],[318,10189,10190],{"xmlns":320,"width":911,"height":912,"viewBox":913,"preserveAspectRatio":914},[327,10191],{"d":917},[19,10193,241],{"className":10194},[240],[19,10196,10198],{"className":10197},[213],[19,10199,10201],{"className":10200,"style":8027},[217],[19,10202],{},[19,10204,241],{"className":10205},[240],[19,10207,10209],{"className":10208},[213],[19,10210,10212],{"className":10211,"style":8039},[217],[19,10213],{},[19,10215],{"className":10216},[674,675],[19,10218,241],{"className":10219},[240],[19,10221,10223],{"className":10222},[213],[19,10224,10227],{"className":10225,"style":10226},[217],"height:0.449em;",[19,10228],{},[19,10230],{"className":10231,"style":2638},[558],[19,10233],{"className":10234,"style":2638},[558],[19,10236,10238],{"className":10237},[2588],[19,10239,10241,10261],{"className":10240},[208,209],[19,10242,10244,10258],{"className":10243},[213],[19,10245,10247],{"className":10246,"style":10089},[217],[19,10248,10249,10252],{"style":10092},[19,10250],{"className":10251,"style":8528},[225],[19,10253,10255],{"className":10254},[195],[19,10256,107],{"className":10257},[195],[19,10259,241],{"className":10260},[240],[19,10262,10264],{"className":10263},[213],[19,10265,10267],{"className":10266,"style":10226},[217],[19,10268],{},[19,10270,10272],{"className":10271,"style":979},[674,978],[19,10273,2471],{"className":10274},[281,10070],[19,10276,10278],{"className":10277},[204],[19,10279,10281],{"className":10280},[208],[19,10282,10284],{"className":10283},[213],[19,10285,10288],{"className":10286,"style":10287},[217],"height:1.3812em;",[19,10289,10291,10294],{"style":10290},"top:-3.6029em;margin-right:0.05em;",[19,10292],{"className":10293,"style":226},[225],[19,10295,10297],{"className":10296},[230,231,232,233],[19,10298,8891],{"className":10299,"style":200},[195,199,233],[19,10301],{"className":10302,"style":971},[254],[19,10304,5791],{"className":10305},[5871],[19,10307],{"className":10308,"style":971},[254],[19,10310,10312,10321],{"className":10311},[195],[19,10313,10315],{"className":10314},[195],[19,10316,10318],{"className":10317},[195],[19,10319,5233],{"className":10320},[195,4792],[19,10322,10324],{"className":10323},[204],[19,10325,10327,10347],{"className":10326},[208,209],[19,10328,10330,10344],{"className":10329},[213],[19,10331,10333],{"className":10332,"style":450},[217],[19,10334,10335,10338],{"style":10021},[19,10336],{"className":10337,"style":226},[225],[19,10339,10341],{"className":10340},[230,231,232,233],[19,10342,158],{"className":10343},[195,233],[19,10345,241],{"className":10346},[240],[19,10348,10350],{"className":10349},[213],[19,10351,10353],{"className":10352,"style":248},[217],[19,10354],{},[19,10356],{"className":10357,"style":255},[254],[19,10359,56],{"className":10360},[259],[19,10362],{"className":10363,"style":255},[254],[19,10365,10367,10370],{"className":10366},[186],[19,10368],{"className":10369,"style":10057},[190],[19,10371,10373,10583],{"className":10372},[273],[19,10374,10376,10382,10577],{"className":10375},[273],[19,10377,10379],{"className":10378,"style":979},[277,978],[19,10380,2436],{"className":10381},[281,10070],[19,10383,10385],{"className":10384},[195],[19,10386,10388,10537,10540,10543],{"className":10387},[64],[19,10389,10391],{"className":10390},[2588],[19,10392,10394,10529],{"className":10393},[208,209],[19,10395,10397,10526],{"className":10396},[213],[19,10398,10400],{"className":10399,"style":10089},[217],[19,10401,10402,10405],{"style":10092},[19,10403],{"className":10404,"style":8528},[225],[19,10406,10408],{"className":10407},[195],[19,10409,10411,10414,10523],{"className":10410},[195],[19,10412],{"className":10413},[277,675],[19,10415,10417],{"className":10416},[703],[19,10418,10420,10515],{"className":10419},[208,209],[19,10421,10423,10512],{"className":10422},[213],[19,10424,10426,10440,10448],{"className":10425,"style":7938},[217],[19,10427,10428,10431],{"style":7941},[19,10429],{"className":10430,"style":865},[225],[19,10432,10434],{"className":10433},[230,231,232,233],[19,10435,10437],{"className":10436},[195,233],[19,10438,158],{"className":10439},[195,233],[19,10441,10442,10445],{"style":932},[19,10443],{"className":10444,"style":865},[225],[19,10446],{"className":10447,"style":940},[939],[19,10449,10450,10453],{"style":7964},[19,10451],{"className":10452,"style":865},[225],[19,10454,10456],{"className":10455},[230,231,232,233],[19,10457,10459,10462,10465],{"className":10458},[195,233],[19,10460,107],{"className":10461},[195,233],[19,10463,140],{"className":10464},[468,233],[19,10466,10468],{"className":10467},[195,872,233],[19,10469,10471,10504],{"className":10470},[208,209],[19,10472,10474,10501],{"className":10473},[213],[19,10475,10477,10489],{"className":10476,"style":7992},[217],[19,10478,10480,10483],{"className":10479,"style":887},[886],[19,10481],{"className":10482,"style":865},[225],[19,10484,10486],{"className":10485,"style":894},[195,233],[19,10487,712],{"className":10488},[195,233],[19,10490,10491,10494],{"style":8007},[19,10492],{"className":10493,"style":865},[225],[19,10495,10497],{"className":10496,"style":908},[907,233],[318,10498,10499],{"xmlns":320,"width":911,"height":912,"viewBox":913,"preserveAspectRatio":914},[327,10500],{"d":917},[19,10502,241],{"className":10503},[240],[19,10505,10507],{"className":10506},[213],[19,10508,10510],{"className":10509,"style":8027},[217],[19,10511],{},[19,10513,241],{"className":10514},[240],[19,10516,10518],{"className":10517},[213],[19,10519,10521],{"className":10520,"style":8039},[217],[19,10522],{},[19,10524],{"className":10525},[674,675],[19,10527,241],{"className":10528},[240],[19,10530,10532],{"className":10531},[213],[19,10533,10535],{"className":10534,"style":10226},[217],[19,10536],{},[19,10538],{"className":10539,"style":2638},[558],[19,10541],{"className":10542,"style":2638},[558],[19,10544,10546],{"className":10545},[2588],[19,10547,10549,10569],{"className":10548},[208,209],[19,10550,10552,10566],{"className":10551},[213],[19,10553,10555],{"className":10554,"style":10089},[217],[19,10556,10557,10560],{"style":10092},[19,10558],{"className":10559,"style":8528},[225],[19,10561,10563],{"className":10562},[195],[19,10564,107],{"className":10565},[195],[19,10567,241],{"className":10568},[240],[19,10570,10572],{"className":10571},[213],[19,10573,10575],{"className":10574,"style":10226},[217],[19,10576],{},[19,10578,10580],{"className":10579,"style":979},[674,978],[19,10581,2471],{"className":10582},[281,10070],[19,10584,10586],{"className":10585},[204],[19,10587,10589],{"className":10588},[208],[19,10590,10592],{"className":10591},[213],[19,10593,10595],{"className":10594,"style":10287},[217],[19,10596,10597,10600],{"style":10290},[19,10598],{"className":10599,"style":226},[225],[19,10601,10603],{"className":10602},[230,231,232,233],[19,10604,8891],{"className":10605,"style":200},[195,199,233],[10,10607,10608,10609,10643],{},"So, the invertible matrix ",[19,10610,10612,10625],{"className":10611,"translate":23},[27],[19,10613,10615],{"className":10614},[31],[33,10616,10617],{"xmlns":35},[38,10618,10619,10623],{},[41,10620,10621],{},[47,10622,5933],{"mathvariant":4730},[175,10624,5936],{"encoding":177},[19,10626,10628],{"className":10627,"ariaHidden":61},[182],[19,10629,10631,10634],{"className":10630},[186],[19,10632],{"className":10633,"style":4782},[190],[19,10635,10637],{"className":10636},[195],[19,10638,10640],{"className":10639},[195],[19,10641,5933],{"className":10642,"style":5955},[195,4792]," is formed by these two eigenvectors:",[19,10645,10647],{"className":10646,"translate":23},[22],[19,10648,10650,10726],{"className":10649,"translate":23},[27],[19,10651,10653],{"className":10652},[31],[33,10654,10655],{"xmlns":35,"display":36},[38,10656,10657,10723],{},[41,10658,10659,10661,10663],{},[47,10660,5933],{"mathvariant":4730},[54,10662,56],{},[41,10664,10665,10667,10721],{},[54,10666,2436],{"fence":61},[64,10668,10669,10707],{"rowspacing":2439,"columnalign":2440,"columnspacing":68},[70,10670,10671,10689],{},[73,10672,10673],{},[76,10674,10675],{"scriptlevel":78,"displaystyle":79},[703,10676,10677,10687],{},[41,10678,10679,10681,10683],{},[81,10680,107],{},[54,10682,145],{},[708,10684,10685],{},[81,10686,712],{},[81,10688,158],{},[73,10690,10691],{},[76,10692,10693],{"scriptlevel":78,"displaystyle":79},[703,10694,10695,10705],{},[41,10696,10697,10699,10701],{},[81,10698,107],{},[54,10700,140],{},[708,10702,10703],{},[81,10704,712],{},[81,10706,158],{},[70,10708,10709,10715],{},[73,10710,10711],{},[76,10712,10713],{"scriptlevel":78,"displaystyle":79},[81,10714,107],{},[73,10716,10717],{},[76,10718,10719],{"scriptlevel":78,"displaystyle":79},[81,10720,107],{},[54,10722,2471],{"fence":61},[175,10724,10725],{"encoding":177},"\\boldsymbol{P} = \\begin{bmatrix}\n\\frac{1+\\sqrt{5}}{2} & \\frac{1-\\sqrt{5}}{2} \\\\\n1 & 1\n\\end{bmatrix}",[19,10727,10729,10753],{"className":10728,"ariaHidden":61},[182],[19,10730,10732,10735,10744,10747,10750],{"className":10731},[186],[19,10733],{"className":10734,"style":4782},[190],[19,10736,10738],{"className":10737},[195],[19,10739,10741],{"className":10740},[195],[19,10742,5933],{"className":10743,"style":5955},[195,4792],[19,10745],{"className":10746,"style":255},[254],[19,10748,56],{"className":10749},[259],[19,10751],{"className":10752,"style":255},[254],[19,10754,10756,10760],{"className":10755},[186],[19,10757],{"className":10758,"style":10759},[190],"height:2.598em;vertical-align:-1.049em;",[19,10761,10763,10769,11105],{"className":10762},[273],[19,10764,10766],{"className":10765,"style":979},[277,978],[19,10767,2436],{"className":10768},[281,232],[19,10770,10772],{"className":10771},[195],[19,10773,10775,10939,10942,10945],{"className":10774},[64],[19,10776,10778],{"className":10777},[2588],[19,10779,10781,10930],{"className":10780},[208,209],[19,10782,10784,10927],{"className":10783},[213],[19,10785,10788,10915],{"className":10786,"style":10787},[217],"height:1.549em;",[19,10789,10791,10794],{"style":10790},"top:-3.549em;",[19,10792],{"className":10793,"style":8528},[225],[19,10795,10797],{"className":10796},[195],[19,10798,10800,10803,10912],{"className":10799},[195],[19,10801],{"className":10802},[277,675],[19,10804,10806],{"className":10805},[703],[19,10807,10809,10904],{"className":10808},[208,209],[19,10810,10812,10901],{"className":10811},[213],[19,10813,10815,10829,10837],{"className":10814,"style":7938},[217],[19,10816,10817,10820],{"style":7941},[19,10818],{"className":10819,"style":865},[225],[19,10821,10823],{"className":10822},[230,231,232,233],[19,10824,10826],{"className":10825},[195,233],[19,10827,158],{"className":10828},[195,233],[19,10830,10831,10834],{"style":932},[19,10832],{"className":10833,"style":865},[225],[19,10835],{"className":10836,"style":940},[939],[19,10838,10839,10842],{"style":7964},[19,10840],{"className":10841,"style":865},[225],[19,10843,10845],{"className":10844},[230,231,232,233],[19,10846,10848,10851,10854],{"className":10847},[195,233],[19,10849,107],{"className":10850},[195,233],[19,10852,145],{"className":10853},[468,233],[19,10855,10857],{"className":10856},[195,872,233],[19,10858,10860,10893],{"className":10859},[208,209],[19,10861,10863,10890],{"className":10862},[213],[19,10864,10866,10878],{"className":10865,"style":7992},[217],[19,10867,10869,10872],{"className":10868,"style":887},[886],[19,10870],{"className":10871,"style":865},[225],[19,10873,10875],{"className":10874,"style":894},[195,233],[19,10876,712],{"className":10877},[195,233],[19,10879,10880,10883],{"style":8007},[19,10881],{"className":10882,"style":865},[225],[19,10884,10886],{"className":10885,"style":908},[907,233],[318,10887,10888],{"xmlns":320,"width":911,"height":912,"viewBox":913,"preserveAspectRatio":914},[327,10889],{"d":917},[19,10891,241],{"className":10892},[240],[19,10894,10896],{"className":10895},[213],[19,10897,10899],{"className":10898,"style":8027},[217],[19,10900],{},[19,10902,241],{"className":10903},[240],[19,10905,10907],{"className":10906},[213],[19,10908,10910],{"className":10909,"style":8039},[217],[19,10911],{},[19,10913],{"className":10914},[674,675],[19,10916,10918,10921],{"style":10917},"top:-2.349em;",[19,10919],{"className":10920,"style":8528},[225],[19,10922,10924],{"className":10923},[195],[19,10925,107],{"className":10926},[195],[19,10928,241],{"className":10929},[240],[19,10931,10933],{"className":10932},[213],[19,10934,10937],{"className":10935,"style":10936},[217],"height:1.049em;",[19,10938],{},[19,10940],{"className":10941,"style":2638},[558],[19,10943],{"className":10944,"style":2638},[558],[19,10946,10948],{"className":10947},[2588],[19,10949,10951,11097],{"className":10950},[208,209],[19,10952,10954,11094],{"className":10953},[213],[19,10955,10957,11083],{"className":10956,"style":10787},[217],[19,10958,10959,10962],{"style":10790},[19,10960],{"className":10961,"style":8528},[225],[19,10963,10965],{"className":10964},[195],[19,10966,10968,10971,11080],{"className":10967},[195],[19,10969],{"className":10970},[277,675],[19,10972,10974],{"className":10973},[703],[19,10975,10977,11072],{"className":10976},[208,209],[19,10978,10980,11069],{"className":10979},[213],[19,10981,10983,10997,11005],{"className":10982,"style":7938},[217],[19,10984,10985,10988],{"style":7941},[19,10986],{"className":10987,"style":865},[225],[19,10989,10991],{"className":10990},[230,231,232,233],[19,10992,10994],{"className":10993},[195,233],[19,10995,158],{"className":10996},[195,233],[19,10998,10999,11002],{"style":932},[19,11000],{"className":11001,"style":865},[225],[19,11003],{"className":11004,"style":940},[939],[19,11006,11007,11010],{"style":7964},[19,11008],{"className":11009,"style":865},[225],[19,11011,11013],{"className":11012},[230,231,232,233],[19,11014,11016,11019,11022],{"className":11015},[195,233],[19,11017,107],{"className":11018},[195,233],[19,11020,140],{"className":11021},[468,233],[19,11023,11025],{"className":11024},[195,872,233],[19,11026,11028,11061],{"className":11027},[208,209],[19,11029,11031,11058],{"className":11030},[213],[19,11032,11034,11046],{"className":11033,"style":7992},[217],[19,11035,11037,11040],{"className":11036,"style":887},[886],[19,11038],{"className":11039,"style":865},[225],[19,11041,11043],{"className":11042,"style":894},[195,233],[19,11044,712],{"className":11045},[195,233],[19,11047,11048,11051],{"style":8007},[19,11049],{"className":11050,"style":865},[225],[19,11052,11054],{"className":11053,"style":908},[907,233],[318,11055,11056],{"xmlns":320,"width":911,"height":912,"viewBox":913,"preserveAspectRatio":914},[327,11057],{"d":917},[19,11059,241],{"className":11060},[240],[19,11062,11064],{"className":11063},[213],[19,11065,11067],{"className":11066,"style":8027},[217],[19,11068],{},[19,11070,241],{"className":11071},[240],[19,11073,11075],{"className":11074},[213],[19,11076,11078],{"className":11077,"style":8039},[217],[19,11079],{},[19,11081],{"className":11082},[674,675],[19,11084,11085,11088],{"style":10917},[19,11086],{"className":11087,"style":8528},[225],[19,11089,11091],{"className":11090},[195],[19,11092,107],{"className":11093},[195],[19,11095,241],{"className":11096},[240],[19,11098,11100],{"className":11099},[213],[19,11101,11103],{"className":11102,"style":10936},[217],[19,11104],{},[19,11106,11108],{"className":11107,"style":979},[674,978],[19,11109,2471],{"className":11110},[281,232],[10,11112,11113,11114,5456,11484,11716,11717,11854],{},"Why is it like this? Let ",[19,11115,11117,11185],{"className":11116,"translate":23},[27],[19,11118,11120],{"className":11119},[31],[33,11121,11122],{"xmlns":35},[38,11123,11124,11182],{},[41,11125,11126,11128,11130],{},[47,11127,5933],{"mathvariant":4730},[54,11129,56],{},[41,11131,11132,11134,11180],{},[54,11133,2436],{"fence":61},[64,11135,11136,11158],{"rowspacing":2439,"columnalign":2440,"columnspacing":68},[70,11137,11138,11148],{},[73,11139,11140],{},[76,11141,11142],{"scriptlevel":78,"displaystyle":79},[44,11143,11144,11146],{},[47,11145,5233],{},[81,11147,107],{},[73,11149,11150],{},[76,11151,11152],{"scriptlevel":78,"displaystyle":79},[44,11153,11154,11156],{},[47,11155,5233],{},[81,11157,158],{},[70,11159,11160,11170],{},[73,11161,11162],{},[76,11163,11164],{"scriptlevel":78,"displaystyle":79},[44,11165,11166,11168],{},[47,11167,8886],{},[81,11169,107],{},[73,11171,11172],{},[76,11173,11174],{"scriptlevel":78,"displaystyle":79},[44,11175,11176,11178],{},[47,11177,8886],{},[81,11179,158],{},[54,11181,2471],{"fence":61},[175,11183,11184],{"encoding":177},"\\boldsymbol{P} = \\begin{bmatrix}\nx_1 & x_2 \\\\\ny_1 & y_2\n\\end{bmatrix}",[19,11186,11188,11212],{"className":11187,"ariaHidden":61},[182],[19,11189,11191,11194,11203,11206,11209],{"className":11190},[186],[19,11192],{"className":11193,"style":4782},[190],[19,11195,11197],{"className":11196},[195],[19,11198,11200],{"className":11199},[195],[19,11201,5933],{"className":11202,"style":5955},[195,4792],[19,11204],{"className":11205,"style":255},[254],[19,11207,56],{"className":11208},[259],[19,11210],{"className":11211,"style":255},[254],[19,11213,11215,11218],{"className":11214},[186],[19,11216],{"className":11217,"style":2569},[190],[19,11219,11221,11227,11478],{"className":11220},[273],[19,11222,11224],{"className":11223,"style":979},[277,978],[19,11225,2436],{"className":11226},[281,232],[19,11228,11230],{"className":11229},[195],[19,11231,11233,11353,11356,11359],{"className":11232},[64],[19,11234,11236],{"className":11235},[2588],[19,11237,11239,11345],{"className":11238},[208,209],[19,11240,11242,11342],{"className":11241},[213],[19,11243,11245,11293],{"className":11244,"style":2598},[217],[19,11246,11247,11250],{"style":2601},[19,11248],{"className":11249,"style":865},[225],[19,11251,11253],{"className":11252},[195],[19,11254,11256,11259],{"className":11255},[195],[19,11257,5233],{"className":11258},[195,199],[19,11260,11262],{"className":11261},[204],[19,11263,11265,11285],{"className":11264},[208,209],[19,11266,11268,11282],{"className":11267},[213],[19,11269,11271],{"className":11270,"style":450},[217],[19,11272,11273,11276],{"style":5603},[19,11274],{"className":11275,"style":226},[225],[19,11277,11279],{"className":11278},[230,231,232,233],[19,11280,107],{"className":11281},[195,233],[19,11283,241],{"className":11284},[240],[19,11286,11288],{"className":11287},[213],[19,11289,11291],{"className":11290,"style":248},[217],[19,11292],{},[19,11294,11295,11298],{"style":2613},[19,11296],{"className":11297,"style":865},[225],[19,11299,11301],{"className":11300},[195],[19,11302,11304,11307],{"className":11303},[195],[19,11305,8886],{"className":11306,"style":9015},[195,199],[19,11308,11310],{"className":11309},[204],[19,11311,11313,11334],{"className":11312},[208,209],[19,11314,11316,11331],{"className":11315},[213],[19,11317,11319],{"className":11318,"style":450},[217],[19,11320,11322,11325],{"style":11321},"top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;",[19,11323],{"className":11324,"style":226},[225],[19,11326,11328],{"className":11327},[230,231,232,233],[19,11329,107],{"className":11330},[195,233],[19,11332,241],{"className":11333},[240],[19,11335,11337],{"className":11336},[213],[19,11338,11340],{"className":11339,"style":248},[217],[19,11341],{},[19,11343,241],{"className":11344},[240],[19,11346,11348],{"className":11347},[213],[19,11349,11351],{"className":11350,"style":2632},[217],[19,11352],{},[19,11354],{"className":11355,"style":2638},[558],[19,11357],{"className":11358,"style":2638},[558],[19,11360,11362],{"className":11361},[2588],[19,11363,11365,11470],{"className":11364},[208,209],[19,11366,11368,11467],{"className":11367},[213],[19,11369,11371,11419],{"className":11370,"style":2598},[217],[19,11372,11373,11376],{"style":2601},[19,11374],{"className":11375,"style":865},[225],[19,11377,11379],{"className":11378},[195],[19,11380,11382,11385],{"className":11381},[195],[19,11383,5233],{"className":11384},[195,199],[19,11386,11388],{"className":11387},[204],[19,11389,11391,11411],{"className":11390},[208,209],[19,11392,11394,11408],{"className":11393},[213],[19,11395,11397],{"className":11396,"style":450},[217],[19,11398,11399,11402],{"style":5603},[19,11400],{"className":11401,"style":226},[225],[19,11403,11405],{"className":11404},[230,231,232,233],[19,11406,158],{"className":11407},[195,233],[19,11409,241],{"className":11410},[240],[19,11412,11414],{"className":11413},[213],[19,11415,11417],{"className":11416,"style":248},[217],[19,11418],{},[19,11420,11421,11424],{"style":2613},[19,11422],{"className":11423,"style":865},[225],[19,11425,11427],{"className":11426},[195],[19,11428,11430,11433],{"className":11429},[195],[19,11431,8886],{"className":11432,"style":9015},[195,199],[19,11434,11436],{"className":11435},[204],[19,11437,11439,11459],{"className":11438},[208,209],[19,11440,11442,11456],{"className":11441},[213],[19,11443,11445],{"className":11444,"style":450},[217],[19,11446,11447,11450],{"style":11321},[19,11448],{"className":11449,"style":226},[225],[19,11451,11453],{"className":11452},[230,231,232,233],[19,11454,158],{"className":11455},[195,233],[19,11457,241],{"className":11458},[240],[19,11460,11462],{"className":11461},[213],[19,11463,11465],{"className":11464,"style":248},[217],[19,11466],{},[19,11468,241],{"className":11469},[240],[19,11471,11473],{"className":11472},[213],[19,11474,11476],{"className":11475,"style":2632},[217],[19,11477],{},[19,11479,11481],{"className":11480,"style":979},[674,978],[19,11482,2471],{"className":11483},[281,232],[19,11485,11487,11529],{"className":11486,"translate":23},[27],[19,11488,11490],{"className":11489},[31],[33,11491,11492],{"xmlns":35},[38,11493,11494,11526],{},[41,11495,11496,11502,11504,11510,11512,11518,11520],{},[44,11497,11498,11500],{},[47,11499,5233],{},[81,11501,107],{},[54,11503,5791],{"separator":61},[44,11505,11506,11508],{},[47,11507,8886],{},[81,11509,107],{},[54,11511,5791],{"separator":61},[44,11513,11514,11516],{},[47,11515,5233],{},[81,11517,158],{},[54,11519,5791],{"separator":61},[44,11521,11522,11524],{},[47,11523,8886],{},[81,11525,158],{},[175,11527,11528],{"encoding":177},"x_1, y_1, x_2, y_2",[19,11530,11532],{"className":11531,"ariaHidden":61},[182],[19,11533,11535,11538,11578,11581,11584,11624,11627,11630,11670,11673,11676],{"className":11534},[186],[19,11536],{"className":11537,"style":9850},[190],[19,11539,11541,11544],{"className":11540},[195],[19,11542,5233],{"className":11543},[195,199],[19,11545,11547],{"className":11546},[204],[19,11548,11550,11570],{"className":11549},[208,209],[19,11551,11553,11567],{"className":11552},[213],[19,11554,11556],{"className":11555,"style":450},[217],[19,11557,11558,11561],{"style":5603},[19,11559],{"className":11560,"style":226},[225],[19,11562,11564],{"className":11563},[230,231,232,233],[19,11565,107],{"className":11566},[195,233],[19,11568,241],{"className":11569},[240],[19,11571,11573],{"className":11572},[213],[19,11574,11576],{"className":11575,"style":248},[217],[19,11577],{},[19,11579,5791],{"className":11580},[5871],[19,11582],{"className":11583,"style":971},[254],[19,11585,11587,11590],{"className":11586},[195],[19,11588,8886],{"className":11589,"style":9015},[195,199],[19,11591,11593],{"className":11592},[204],[19,11594,11596,11616],{"className":11595},[208,209],[19,11597,11599,11613],{"className":11598},[213],[19,11600,11602],{"className":11601,"style":450},[217],[19,11603,11604,11607],{"style":11321},[19,11605],{"className":11606,"style":226},[225],[19,11608,11610],{"className":11609},[230,231,232,233],[19,11611,107],{"className":11612},[195,233],[19,11614,241],{"className":11615},[240],[19,11617,11619],{"className":11618},[213],[19,11620,11622],{"className":11621,"style":248},[217],[19,11623],{},[19,11625,5791],{"className":11626},[5871],[19,11628],{"className":11629,"style":971},[254],[19,11631,11633,11636],{"className":11632},[195],[19,11634,5233],{"className":11635},[195,199],[19,11637,11639],{"className":11638},[204],[19,11640,11642,11662],{"className":11641},[208,209],[19,11643,11645,11659],{"className":11644},[213],[19,11646,11648],{"className":11647,"style":450},[217],[19,11649,11650,11653],{"style":5603},[19,11651],{"className":11652,"style":226},[225],[19,11654,11656],{"className":11655},[230,231,232,233],[19,11657,158],{"className":11658},[195,233],[19,11660,241],{"className":11661},[240],[19,11663,11665],{"className":11664},[213],[19,11666,11668],{"className":11667,"style":248},[217],[19,11669],{},[19,11671,5791],{"className":11672},[5871],[19,11674],{"className":11675,"style":971},[254],[19,11677,11679,11682],{"className":11678},[195],[19,11680,8886],{"className":11681,"style":9015},[195,199],[19,11683,11685],{"className":11684},[204],[19,11686,11688,11708],{"className":11687},[208,209],[19,11689,11691,11705],{"className":11690},[213],[19,11692,11694],{"className":11693,"style":450},[217],[19,11695,11696,11699],{"style":11321},[19,11697],{"className":11698,"style":226},[225],[19,11700,11702],{"className":11701},[230,231,232,233],[19,11703,158],{"className":11704},[195,233],[19,11706,241],{"className":11707},[240],[19,11709,11711],{"className":11710},[213],[19,11712,11714],{"className":11713,"style":248},[217],[19,11715],{}," are the components of eigenvectors ",[19,11718,11720,11746],{"className":11719,"translate":23},[27],[19,11721,11723],{"className":11722},[31],[33,11724,11725],{"xmlns":35},[38,11726,11727,11743],{},[41,11728,11729,11735,11737],{},[44,11730,11731,11733],{},[47,11732,5233],{"mathvariant":4730},[81,11734,107],{},[54,11736,5791],{"separator":61},[44,11738,11739,11741],{},[47,11740,5233],{"mathvariant":4730},[81,11742,158],{},[175,11744,11745],{"encoding":177},"\\boldsymbol{x}_1, \\boldsymbol{x}_2",[19,11747,11749],{"className":11748,"ariaHidden":61},[182],[19,11750,11752,11756,11802,11805,11808],{"className":11751},[186],[19,11753],{"className":11754,"style":11755},[190],"height:0.6389em;vertical-align:-0.1944em;",[19,11757,11759,11768],{"className":11758},[195],[19,11760,11762],{"className":11761},[195],[19,11763,11765],{"className":11764},[195],[19,11766,5233],{"className":11767},[195,4792],[19,11769,11771],{"className":11770},[204],[19,11772,11774,11794],{"className":11773},[208,209],[19,11775,11777,11791],{"className":11776},[213],[19,11778,11780],{"className":11779,"style":450},[217],[19,11781,11782,11785],{"style":10021},[19,11783],{"className":11784,"style":226},[225],[19,11786,11788],{"className":11787},[230,231,232,233],[19,11789,107],{"className":11790},[195,233],[19,11792,241],{"className":11793},[240],[19,11795,11797],{"className":11796},[213],[19,11798,11800],{"className":11799,"style":248},[217],[19,11801],{},[19,11803,5791],{"className":11804},[5871],[19,11806],{"className":11807,"style":971},[254],[19,11809,11811,11820],{"className":11810},[195],[19,11812,11814],{"className":11813},[195],[19,11815,11817],{"className":11816},[195],[19,11818,5233],{"className":11819},[195,4792],[19,11821,11823],{"className":11822},[204],[19,11824,11826,11846],{"className":11825},[208,209],[19,11827,11829,11843],{"className":11828},[213],[19,11830,11832],{"className":11831,"style":450},[217],[19,11833,11834,11837],{"style":10021},[19,11835],{"className":11836,"style":226},[225],[19,11838,11840],{"className":11839},[230,231,232,233],[19,11841,158],{"className":11842},[195,233],[19,11844,241],{"className":11845},[240],[19,11847,11849],{"className":11848},[213],[19,11850,11852],{"className":11851,"style":248},[217],[19,11853],{}," respectively).",[10,11856,11857,11858,12024,12025,12556,12557,12762,12763,12838,12839,13083,13084,13452],{},"Now, if we calculate ",[19,11859,11861,11905],{"className":11860,"translate":23},[27],[19,11862,11864],{"className":11863},[31],[33,11865,11866],{"xmlns":35},[38,11867,11868,11902],{},[41,11869,11870,11872],{},[47,11871,5933],{"mathvariant":4730},[41,11873,11874,11884,11886,11892,11894,11900],{},[41,11875,11876,11878,11880,11882],{},[47,11877,5771],{"mathvariant":5770},[47,11879,5774],{"mathvariant":5770},[47,11881,5777],{"mathvariant":5770},[47,11883,5780],{"mathvariant":5770},[54,11885,62],{"stretchy":79},[44,11887,11888,11890],{},[47,11889,5238],{},[81,11891,107],{},[54,11893,5791],{"separator":61},[44,11895,11896,11898],{},[47,11897,5238],{},[81,11899,158],{},[54,11901,5800],{"stretchy":79},[175,11903,11904],{"encoding":177},"\\boldsymbol{P}\\rm{diag}\\{\\lambda_1, \\lambda_2\\}",[19,11906,11908],{"className":11907,"ariaHidden":61},[182],[19,11909,11911,11914,11923],{"className":11910},[186],[19,11912],{"className":11913,"style":5312},[190],[19,11915,11917],{"className":11916},[195],[19,11918,11920],{"className":11919},[195],[19,11921,5933],{"className":11922,"style":5955},[195,4792],[19,11924,11926,11932,11935,11975,11978,11981,12021],{"className":11925},[195],[19,11927,11929],{"className":11928},[195],[19,11930,5824],{"className":11931,"style":5823},[195,5822],[19,11933,62],{"className":11934},[277],[19,11936,11938,11941],{"className":11937},[195],[19,11939,5238],{"className":11940},[195,199],[19,11942,11944],{"className":11943},[204],[19,11945,11947,11967],{"className":11946},[208,209],[19,11948,11950,11964],{"className":11949},[213],[19,11951,11953],{"className":11952,"style":450},[217],[19,11954,11955,11958],{"style":5603},[19,11956],{"className":11957,"style":226},[225],[19,11959,11961],{"className":11960},[230,231,232,233],[19,11962,107],{"className":11963},[195,5822,233],[19,11965,241],{"className":11966},[240],[19,11968,11970],{"className":11969},[213],[19,11971,11973],{"className":11972,"style":248},[217],[19,11974],{},[19,11976,5791],{"className":11977},[5871],[19,11979],{"className":11980,"style":971},[254],[19,11982,11984,11987],{"className":11983},[195],[19,11985,5238],{"className":11986},[195,199],[19,11988,11990],{"className":11989},[204],[19,11991,11993,12013],{"className":11992},[208,209],[19,11994,11996,12010],{"className":11995},[213],[19,11997,11999],{"className":11998,"style":450},[217],[19,12000,12001,12004],{"style":5603},[19,12002],{"className":12003,"style":226},[225],[19,12005,12007],{"className":12006},[230,231,232,233],[19,12008,158],{"className":12009},[195,5822,233],[19,12011,241],{"className":12012},[240],[19,12014,12016],{"className":12015},[213],[19,12017,12019],{"className":12018,"style":248},[217],[19,12020],{},[19,12022,5800],{"className":12023},[674],", it exactly equals ",[19,12026,12028,12122],{"className":12027,"translate":23},[27],[19,12029,12031],{"className":12030},[31],[33,12032,12033],{"xmlns":35},[38,12034,12035,12119],{},[41,12036,12037,12039,12117],{},[54,12038,2436],{"fence":61},[64,12040,12041,12079],{"rowspacing":2439,"columnalign":2440,"columnspacing":68},[70,12042,12043,12061],{},[73,12044,12045],{},[76,12046,12047],{"scriptlevel":78,"displaystyle":79},[41,12048,12049,12055],{},[44,12050,12051,12053],{},[47,12052,5238],{},[81,12054,107],{},[44,12056,12057,12059],{},[47,12058,5233],{},[81,12060,107],{},[73,12062,12063],{},[76,12064,12065],{"scriptlevel":78,"displaystyle":79},[41,12066,12067,12073],{},[44,12068,12069,12071],{},[47,12070,5238],{},[81,12072,158],{},[44,12074,12075,12077],{},[47,12076,5233],{},[81,12078,158],{},[70,12080,12081,12099],{},[73,12082,12083],{},[76,12084,12085],{"scriptlevel":78,"displaystyle":79},[41,12086,12087,12093],{},[44,12088,12089,12091],{},[47,12090,5238],{},[81,12092,107],{},[44,12094,12095,12097],{},[47,12096,8886],{},[81,12098,107],{},[73,12100,12101],{},[76,12102,12103],{"scriptlevel":78,"displaystyle":79},[41,12104,12105,12111],{},[44,12106,12107,12109],{},[47,12108,5238],{},[81,12110,158],{},[44,12112,12113,12115],{},[47,12114,8886],{},[81,12116,158],{},[54,12118,2471],{"fence":61},[175,12120,12121],{"encoding":177},"\\begin{bmatrix}\n\\lambda_1x_1 & \\lambda_2x_2 \\\\\n\\lambda_1y_1 & \\lambda_2y_2\n\\end{bmatrix}",[19,12123,12125],{"className":12124,"ariaHidden":61},[182],[19,12126,12128,12131],{"className":12127},[186],[19,12129],{"className":12130,"style":2569},[190],[19,12132,12134,12140,12550],{"className":12133},[273],[19,12135,12137],{"className":12136,"style":979},[277,978],[19,12138,2436],{"className":12139},[281,232],[19,12141,12143],{"className":12142},[195],[19,12144,12146,12345,12348,12351],{"className":12145},[64],[19,12147,12149],{"className":12148},[2588],[19,12150,12152,12337],{"className":12151},[208,209],[19,12153,12155,12334],{"className":12154},[213],[19,12156,12158,12246],{"className":12157,"style":2598},[217],[19,12159,12160,12163],{"style":2601},[19,12161],{"className":12162,"style":865},[225],[19,12164,12166,12206],{"className":12165},[195],[19,12167,12169,12172],{"className":12168},[195],[19,12170,5238],{"className":12171},[195,199],[19,12173,12175],{"className":12174},[204],[19,12176,12178,12198],{"className":12177},[208,209],[19,12179,12181,12195],{"className":12180},[213],[19,12182,12184],{"className":12183,"style":450},[217],[19,12185,12186,12189],{"style":5603},[19,12187],{"className":12188,"style":226},[225],[19,12190,12192],{"className":12191},[230,231,232,233],[19,12193,107],{"className":12194},[195,233],[19,12196,241],{"className":12197},[240],[19,12199,12201],{"className":12200},[213],[19,12202,12204],{"className":12203,"style":248},[217],[19,12205],{},[19,12207,12209,12212],{"className":12208},[195],[19,12210,5233],{"className":12211},[195,199],[19,12213,12215],{"className":12214},[204],[19,12216,12218,12238],{"className":12217},[208,209],[19,12219,12221,12235],{"className":12220},[213],[19,12222,12224],{"className":12223,"style":450},[217],[19,12225,12226,12229],{"style":5603},[19,12227],{"className":12228,"style":226},[225],[19,12230,12232],{"className":12231},[230,231,232,233],[19,12233,107],{"className":12234},[195,233],[19,12236,241],{"className":12237},[240],[19,12239,12241],{"className":12240},[213],[19,12242,12244],{"className":12243,"style":248},[217],[19,12245],{},[19,12247,12248,12251],{"style":2613},[19,12249],{"className":12250,"style":865},[225],[19,12252,12254,12294],{"className":12253},[195],[19,12255,12257,12260],{"className":12256},[195],[19,12258,5238],{"className":12259},[195,199],[19,12261,12263],{"className":12262},[204],[19,12264,12266,12286],{"className":12265},[208,209],[19,12267,12269,12283],{"className":12268},[213],[19,12270,12272],{"className":12271,"style":450},[217],[19,12273,12274,12277],{"style":5603},[19,12275],{"className":12276,"style":226},[225],[19,12278,12280],{"className":12279},[230,231,232,233],[19,12281,107],{"className":12282},[195,233],[19,12284,241],{"className":12285},[240],[19,12287,12289],{"className":12288},[213],[19,12290,12292],{"className":12291,"style":248},[217],[19,12293],{},[19,12295,12297,12300],{"className":12296},[195],[19,12298,8886],{"className":12299,"style":9015},[195,199],[19,12301,12303],{"className":12302},[204],[19,12304,12306,12326],{"className":12305},[208,209],[19,12307,12309,12323],{"className":12308},[213],[19,12310,12312],{"className":12311,"style":450},[217],[19,12313,12314,12317],{"style":11321},[19,12315],{"className":12316,"style":226},[225],[19,12318,12320],{"className":12319},[230,231,232,233],[19,12321,107],{"className":12322},[195,233],[19,12324,241],{"className":12325},[240],[19,12327,12329],{"className":12328},[213],[19,12330,12332],{"className":12331,"style":248},[217],[19,12333],{},[19,12335,241],{"className":12336},[240],[19,12338,12340],{"className":12339},[213],[19,12341,12343],{"className":12342,"style":2632},[217],[19,12344],{},[19,12346],{"className":12347,"style":2638},[558],[19,12349],{"className":12350,"style":2638},[558],[19,12352,12354],{"className":12353},[2588],[19,12355,12357,12542],{"className":12356},[208,209],[19,12358,12360,12539],{"className":12359},[213],[19,12361,12363,12451],{"className":12362,"style":2598},[217],[19,12364,12365,12368],{"style":2601},[19,12366],{"className":12367,"style":865},[225],[19,12369,12371,12411],{"className":12370},[195],[19,12372,12374,12377],{"className":12373},[195],[19,12375,5238],{"className":12376},[195,199],[19,12378,12380],{"className":12379},[204],[19,12381,12383,12403],{"className":12382},[208,209],[19,12384,12386,12400],{"className":12385},[213],[19,12387,12389],{"className":12388,"style":450},[217],[19,12390,12391,12394],{"style":5603},[19,12392],{"className":12393,"style":226},[225],[19,12395,12397],{"className":12396},[230,231,232,233],[19,12398,158],{"className":12399},[195,233],[19,12401,241],{"className":12402},[240],[19,12404,12406],{"className":12405},[213],[19,12407,12409],{"className":12408,"style":248},[217],[19,12410],{},[19,12412,12414,12417],{"className":12413},[195],[19,12415,5233],{"className":12416},[195,199],[19,12418,12420],{"className":12419},[204],[19,12421,12423,12443],{"className":12422},[208,209],[19,12424,12426,12440],{"className":12425},[213],[19,12427,12429],{"className":12428,"style":450},[217],[19,12430,12431,12434],{"style":5603},[19,12432],{"className":12433,"style":226},[225],[19,12435,12437],{"className":12436},[230,231,232,233],[19,12438,158],{"className":12439},[195,233],[19,12441,241],{"className":12442},[240],[19,12444,12446],{"className":12445},[213],[19,12447,12449],{"className":12448,"style":248},[217],[19,12450],{},[19,12452,12453,12456],{"style":2613},[19,12454],{"className":12455,"style":865},[225],[19,12457,12459,12499],{"className":12458},[195],[19,12460,12462,12465],{"className":12461},[195],[19,12463,5238],{"className":12464},[195,199],[19,12466,12468],{"className":12467},[204],[19,12469,12471,12491],{"className":12470},[208,209],[19,12472,12474,12488],{"className":12473},[213],[19,12475,12477],{"className":12476,"style":450},[217],[19,12478,12479,12482],{"style":5603},[19,12480],{"className":12481,"style":226},[225],[19,12483,12485],{"className":12484},[230,231,232,233],[19,12486,158],{"className":12487},[195,233],[19,12489,241],{"className":12490},[240],[19,12492,12494],{"className":12493},[213],[19,12495,12497],{"className":12496,"style":248},[217],[19,12498],{},[19,12500,12502,12505],{"className":12501},[195],[19,12503,8886],{"className":12504,"style":9015},[195,199],[19,12506,12508],{"className":12507},[204],[19,12509,12511,12531],{"className":12510},[208,209],[19,12512,12514,12528],{"className":12513},[213],[19,12515,12517],{"className":12516,"style":450},[217],[19,12518,12519,12522],{"style":11321},[19,12520],{"className":12521,"style":226},[225],[19,12523,12525],{"className":12524},[230,231,232,233],[19,12526,158],{"className":12527},[195,233],[19,12529,241],{"className":12530},[240],[19,12532,12534],{"className":12533},[213],[19,12535,12537],{"className":12536,"style":248},[217],[19,12538],{},[19,12540,241],{"className":12541},[240],[19,12543,12545],{"className":12544},[213],[19,12546,12548],{"className":12547,"style":2632},[217],[19,12549],{},[19,12551,12553],{"className":12552,"style":979},[674,978],[19,12554,2471],{"className":12555},[281,232],". This means that ",[19,12558,12560,12610],{"className":12559,"translate":23},[27],[19,12561,12563],{"className":12562},[31],[33,12564,12565],{"xmlns":35},[38,12566,12567,12607],{},[41,12568,12569,12571,12573,12575,12577],{},[47,12570,4731],{"mathvariant":4730},[47,12572,5933],{"mathvariant":4730},[54,12574,56],{},[47,12576,5933],{"mathvariant":4730},[41,12578,12579,12589,12591,12597,12599,12605],{},[41,12580,12581,12583,12585,12587],{},[47,12582,5771],{"mathvariant":5770},[47,12584,5774],{"mathvariant":5770},[47,12586,5777],{"mathvariant":5770},[47,12588,5780],{"mathvariant":5770},[54,12590,62],{"stretchy":79},[44,12592,12593,12595],{},[47,12594,5238],{},[81,12596,107],{},[54,12598,5791],{"separator":61},[44,12600,12601,12603],{},[47,12602,5238],{},[81,12604,158],{},[54,12606,5800],{"stretchy":79},[175,12608,12609],{"encoding":177},"\\boldsymbol{A}\\boldsymbol{P} = \\boldsymbol{P}\\rm{diag}\\{\\lambda_1, \\lambda_2\\}",[19,12611,12613,12646],{"className":12612,"ariaHidden":61},[182],[19,12614,12616,12619,12628,12637,12640,12643],{"className":12615},[186],[19,12617],{"className":12618,"style":4782},[190],[19,12620,12622],{"className":12621},[195],[19,12623,12625],{"className":12624},[195],[19,12626,4731],{"className":12627},[195,4792],[19,12629,12631],{"className":12630},[195],[19,12632,12634],{"className":12633},[195],[19,12635,5933],{"className":12636,"style":5955},[195,4792],[19,12638],{"className":12639,"style":255},[254],[19,12641,56],{"className":12642},[259],[19,12644],{"className":12645,"style":255},[254],[19,12647,12649,12652,12661],{"className":12648},[186],[19,12650],{"className":12651,"style":5312},[190],[19,12653,12655],{"className":12654},[195],[19,12656,12658],{"className":12657},[195],[19,12659,5933],{"className":12660,"style":5955},[195,4792],[19,12662,12664,12670,12673,12713,12716,12719,12759],{"className":12663},[195],[19,12665,12667],{"className":12666},[195],[19,12668,5824],{"className":12669,"style":5823},[195,5822],[19,12671,62],{"className":12672},[277],[19,12674,12676,12679],{"className":12675},[195],[19,12677,5238],{"className":12678},[195,199],[19,12680,12682],{"className":12681},[204],[19,12683,12685,12705],{"className":12684},[208,209],[19,12686,12688,12702],{"className":12687},[213],[19,12689,12691],{"className":12690,"style":450},[217],[19,12692,12693,12696],{"style":5603},[19,12694],{"className":12695,"style":226},[225],[19,12697,12699],{"className":12698},[230,231,232,233],[19,12700,107],{"className":12701},[195,5822,233],[19,12703,241],{"className":12704},[240],[19,12706,12708],{"className":12707},[213],[19,12709,12711],{"className":12710,"style":248},[217],[19,12712],{},[19,12714,5791],{"className":12715},[5871],[19,12717],{"className":12718,"style":971},[254],[19,12720,12722,12725],{"className":12721},[195],[19,12723,5238],{"className":12724},[195,199],[19,12726,12728],{"className":12727},[204],[19,12729,12731,12751],{"className":12730},[208,209],[19,12732,12734,12748],{"className":12733},[213],[19,12735,12737],{"className":12736,"style":450},[217],[19,12738,12739,12742],{"style":5603},[19,12740],{"className":12741,"style":226},[225],[19,12743,12745],{"className":12744},[230,231,232,233],[19,12746,158],{"className":12747},[195,5822,233],[19,12749,241],{"className":12750},[240],[19,12752,12754],{"className":12753},[213],[19,12755,12757],{"className":12756,"style":248},[217],[19,12758],{},[19,12760,5800],{"className":12761},[674]," is inevitable. Left-multiplying by ",[19,12764,12766,12788],{"className":12765,"translate":23},[27],[19,12767,12769],{"className":12768},[31],[33,12770,12771],{"xmlns":35},[38,12772,12773,12785],{},[41,12774,12775],{},[719,12776,12777,12779],{},[47,12778,5933],{"mathvariant":4730},[41,12780,12781,12783],{},[54,12782,140],{},[81,12784,107],{},[175,12786,12787],{"encoding":177},"\\boldsymbol{P}^{-1}",[19,12789,12791],{"className":12790,"ariaHidden":61},[182],[19,12792,12794,12797],{"className":12793},[186],[19,12795],{"className":12796,"style":4962},[190],[19,12798,12800,12809],{"className":12799},[195],[19,12801,12803],{"className":12802},[195],[19,12804,12806],{"className":12805},[195],[19,12807,5933],{"className":12808,"style":5955},[195,4792],[19,12810,12812],{"className":12811},[204],[19,12813,12815],{"className":12814},[208],[19,12816,12818],{"className":12817},[213],[19,12819,12821],{"className":12820,"style":4962},[217],[19,12822,12823,12826],{"style":4989},[19,12824],{"className":12825,"style":226},[225],[19,12827,12829],{"className":12828},[230,231,232,233],[19,12830,12832,12835],{"className":12831},[195,233],[19,12833,140],{"className":12834},[195,233],[19,12836,107],{"className":12837},[195,233],", we get 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So, 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is reasonable.",[10,13454,13455],{},"Its inverse matrix is easy to 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= \\frac{\\boldsymbol{P}^*}{\\left|\\boldsymbol{P}\\right|} = \\frac{1}{\\sqrt{5}}\\begin{bmatrix}\n1 & \\frac{1-\\sqrt{5}}{2} \\\\\n1 & 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],[19,14166,712],{"className":14167},[195,233],[19,14169,14170,14173],{"style":8007},[19,14171],{"className":14172,"style":865},[225],[19,14174,14176],{"className":14175,"style":908},[907,233],[318,14177,14178],{"xmlns":320,"width":911,"height":912,"viewBox":913,"preserveAspectRatio":914},[327,14179],{"d":917},[19,14181,241],{"className":14182},[240],[19,14184,14186],{"className":14185},[213],[19,14187,14189],{"className":14188,"style":8027},[217],[19,14190],{},[19,14192,241],{"className":14193},[240],[19,14195,14197],{"className":14196},[213],[19,14198,14200],{"className":14199,"style":8039},[217],[19,14201],{},[19,14203],{"className":14204},[674,675],[19,14206,241],{"className":14207},[240],[19,14209,14211],{"className":14210},[213],[19,14212,14214],{"className":14213,"style":8670},[217],[19,14215],{},[19,14217,14219],{"className":14218,"style":979},[674,978],[19,14220,2471],{"className":14221},[281,983],[10,14223,14224,14225,14263],{},"Substituting into equation ",[19,14226,14228,14245],{"className":14227,"translate":23},[27],[19,14229,14231],{"className":14230},[31],[33,14232,14233],{"xmlns":35},[38,14234,14235,14243],{},[41,14236,14237,14239,14241],{},[54,14238,717],{"stretchy":79},[81,14240,158],{},[54,14242,741],{"stretchy":79},[175,14244,6333],{"encoding":177},[19,14246,14248],{"className":14247,"ariaHidden":61},[182],[19,14249,14251,14254,14257,14260],{"className":14250},[186],[19,14252],{"className":14253,"style":5312},[190],[19,14255,717],{"className":14256},[277],[19,14258,158],{"className":14259},[195],[19,14261,741],{"className":14262},[674],":",[19,14265,14267],{"className":14266,"translate":23},[22],[19,14268,14270,14372],{"className":14269,"translate":23},[27],[19,14271,14273],{"className":14272},[31],[33,14274,14275],{"xmlns":35,"display":36},[38,14276,14277,14369],{},[41,14278,14279,14285,14287,14289,14291,14301,14303,14305,14311,14321,14323,14325],{},[719,14280,14281,14283],{},[47,14282,4731],{"mathvariant":4730},[47,14284,52],{},[54,14286,56],{},[47,14288,5933],{"mathvariant":4730},[54,14290,717],{"stretchy":79},[719,14292,14293,14295],{},[47,14294,5933],{"mathvariant":4730},[41,14296,14297,14299],{},[54,14298,140],{},[81,14300,107],{},[47,14302,4731],{"mathvariant":4730},[47,14304,5933],{"mathvariant":4730},[719,14306,14307,14309],{},[54,14308,741],{"stretchy":79},[47,14310,52],{},[719,14312,14313,14315],{},[47,14314,5933],{"mathvariant":4730},[41,14316,14317,14319],{},[54,14318,140],{},[81,14320,107],{},[54,14322,56],{},[47,14324,5933],{"mathvariant":4730},[41,14326,14327,14341,14343,14349,14351,14357,14359],{},[719,14328,14329,14339],{},[41,14330,14331,14333,14335,14337],{},[47,14332,5771],{"mathvariant":5770},[47,14334,5774],{"mathvariant":5770},[47,14336,5777],{"mathvariant":5770},[47,14338,5780],{"mathvariant":5770},[47,14340,52],{"mathvariant":5770},[54,14342,62],{"stretchy":79},[44,14344,14345,14347],{},[47,14346,5238],{},[81,14348,107],{},[54,14350,5791],{"separator":61},[44,14352,14353,14355],{},[47,14354,5238],{},[81,14356,158],{},[54,14358,5800],{"stretchy":79},[719,14360,14361,14363],{},[47,14362,5933],{"mathvariant":4730},[41,14364,14365,14367],{},[54,14366,140],{},[81,14368,107],{},[175,14370,14371],{"encoding":177},"\\boldsymbol{A}^n = \\boldsymbol{P}(\\boldsymbol{P}^{-1}\\boldsymbol{A}\\boldsymbol{P})^n\\boldsymbol{P}^{-1} = \\boldsymbol{P}\\rm{diag}^n\\{\\lambda_1, \\lambda_2\\}\\boldsymbol{P}^{-1}",[19,14373,14375,14425,14581],{"className":14374,"ariaHidden":61},[182],[19,14376,14378,14381,14416,14419,14422],{"className":14377},[186],[19,14379],{"className":14380,"style":5039},[190],[19,14382,14384,14393],{"className":14383},[195],[19,14385,14387],{"className":14386},[195],[19,14388,14390],{"className":14389},[195],[19,14391,4731],{"className":14392},[195,4792],[19,14394,14396],{"className":14395},[204],[19,14397,14399],{"className":14398},[208],[19,14400,14402],{"className":14401},[213],[19,14403,14405],{"className":14404,"style":5039},[217],[19,14406,14407,14410],{"style":4989},[19,14408],{"className":14409,"style":226},[225],[19,14411,14413],{"className":14412},[230,231,232,233],[19,14414,52],{"className":14415},[195,199,233],[19,14417],{"className":14418,"style":255},[254],[19,14420,56],{"className":14421},[259],[19,14423],{"className":14424,"style":255},[254],[19,14426,14428,14431,14440,14443,14484,14493,14502,14531,14572,14575,14578],{"className":14427},[186],[19,14429],{"className":14430,"style":6346},[190],[19,14432,14434],{"className":14433},[195],[19,14435,14437],{"className":14436},[195],[19,14438,5933],{"className":14439,"style":5955},[195,4792],[19,14441,717],{"className":14442},[277],[19,14444,14446,14455],{"className":14445},[195],[19,14447,14449],{"className":14448},[195],[19,14450,14452],{"className":14451},[195],[19,14453,5933],{"className":14454,"style":5955},[195,4792],[19,14456,14458],{"className":14457},[204],[19,14459,14461],{"className":14460},[208],[19,14462,14464],{"className":14463},[213],[19,14465,14467],{"className":14466,"style":4962},[217],[19,14468,14469,14472],{"style":4989},[19,14470],{"className":14471,"style":226},[225],[19,14473,14475],{"className":14474},[230,231,232,233],[19,14476,14478,14481],{"className":14477},[195,233],[19,14479,140],{"className":14480},[195,233],[19,14482,107],{"className":14483},[195,233],[19,14485,14487],{"className":14486},[195],[19,14488,14490],{"className":14489},[195],[19,14491,4731],{"className":14492},[195,4792],[19,14494,14496],{"className":14495},[195],[19,14497,14499],{"className":14498},[195],[19,14500,5933],{"className":14501,"style":5955},[195,4792],[19,14503,14505,14508],{"className":14504},[674],[19,14506,741],{"className":14507},[674],[19,14509,14511],{"className":14510},[204],[19,14512,14514],{"className":14513},[208],[19,14515,14517],{"className":14516},[213],[19,14518,14520],{"className":14519,"style":6427},[217],[19,14521,14522,14525],{"style":6430},[19,14523],{"className":14524,"style":226},[225],[19,14526,14528],{"className":14527},[230,231,232,233],[19,14529,52],{"className":14530},[195,199,233],[19,14532,14534,14543],{"className":14533},[195],[19,14535,14537],{"className":14536},[195],[19,14538,14540],{"className":14539},[195],[19,14541,5933],{"className":14542,"style":5955},[195,4792],[19,14544,14546],{"className":14545},[204],[19,14547,14549],{"className":14548},[208],[19,14550,14552],{"className":14551},[213],[19,14553,14555],{"className":14554,"style":4962},[217],[19,14556,14557,14560],{"style":4989},[19,14558],{"className":14559,"style":226},[225],[19,14561,14563],{"className":14562},[230,231,232,233],[19,14564,14566,14569],{"className":14565},[195,233],[19,14567,140],{"className":14568},[195,233],[19,14570,107],{"className":14571},[195,233],[19,14573],{"className":14574,"style":255},[254],[19,14576,56],{"className":14577},[259],[19,14579],{"className":14580,"style":255},[254],[19,14582,14584,14587,14596],{"className":14583},[186],[19,14585],{"className":14586,"style":6346},[190],[19,14588,14590],{"className":14589},[195],[19,14591,14593],{"className":14592},[195],[19,14594,5933],{"className":14595,"style":5955},[195,4792],[19,14597,14599,14633,14636,14676,14679,14682,14722,14725],{"className":14598},[195],[19,14600,14602,14608],{"className":14601},[195],[19,14603,14605],{"className":14604},[195],[19,14606,5824],{"className":14607,"style":5823},[195,5822],[19,14609,14611],{"className":14610},[204],[19,14612,14614],{"className":14613},[208],[19,14615,14617],{"className":14616},[213],[19,14618,14621],{"className":14619,"style":14620},[217],"height:0.7487em;",[19,14622,14624,14627],{"style":14623},"top:-3.1473em;margin-right:0.05em;",[19,14625],{"className":14626,"style":226},[225],[19,14628,14630],{"className":14629},[230,231,232,233],[19,14631,52],{"className":14632},[195,5822,233],[19,14634,62],{"className":14635},[277],[19,14637,14639,14642],{"className":14638},[195],[19,14640,5238],{"className":14641},[195,199],[19,14643,14645],{"className":14644},[204],[19,14646,14648,14668],{"className":14647},[208,209],[19,14649,14651,14665],{"className":14650},[213],[19,14652,14654],{"className":14653,"style":450},[217],[19,14655,14656,14659],{"style":5603},[19,14657],{"className":14658,"style":226},[225],[19,14660,14662],{"className":14661},[230,231,232,233],[19,14663,107],{"className":14664},[195,5822,233],[19,14666,241],{"className":14667},[240],[19,14669,14671],{"className":14670},[213],[19,14672,14674],{"className":14673,"style":248},[217],[19,14675],{},[19,14677,5791],{"className":14678},[5871],[19,14680],{"className":14681,"style":971},[254],[19,14683,14685,14688],{"className":14684},[195],[19,14686,5238],{"className":14687},[195,199],[19,14689,14691],{"className":14690},[204],[19,14692,14694,14714],{"className":14693},[208,209],[19,14695,14697,14711],{"className":14696},[213],[19,14698,14700],{"className":14699,"style":450},[217],[19,14701,14702,14705],{"style":5603},[19,14703],{"className":14704,"style":226},[225],[19,14706,14708],{"className":14707},[230,231,232,233],[19,14709,158],{"className":14710},[195,5822,233],[19,14712,241],{"className":14713},[240],[19,14715,14717],{"className":14716},[213],[19,14718,14720],{"className":14719,"style":248},[217],[19,14721],{},[19,14723,5800],{"className":14724},[674],[19,14726,14728,14737],{"className":14727},[195],[19,14729,14731],{"className":14730},[195],[19,14732,14734],{"className":14733},[195],[19,14735,5933],{"className":14736,"style":5955},[195,4792],[19,14738,14740],{"className":14739},[204],[19,14741,14743],{"className":14742},[208],[19,14744,14746],{"className":14745},[213],[19,14747,14749],{"className":14748,"style":4962},[217],[19,14750,14751,14754],{"style":4989},[19,14752],{"className":14753,"style":226},[225],[19,14755,14757],{"className":14756},[230,231,232,233],[19,14758,14760,14763],{"className":14759},[195,233],[19,14761,140],{"className":14762},[195,233],[19,14764,107],{"className":14765},[195,5822,233],[10,14767,14768],{},"Where the diagonal matrix is:",[19,14770,14772],{"className":14771,"translate":23},[22],[19,14773,14775,14901],{"className":14774,"translate":23},[27],[19,14776,14778],{"className":14777},[31],[33,14779,14780],{"xmlns":35,"display":36},[38,14781,14782,14898],{},[41,14783,14784,14798,14800,14806,14808,14814,14816,14818],{},[719,14785,14786,14796],{},[41,14787,14788,14790,14792,14794],{},[47,14789,5771],{"mathvariant":5770},[47,14791,5774],{"mathvariant":5770},[47,14793,5777],{"mathvariant":5770},[47,14795,5780],{"mathvariant":5770},[47,14797,52],{"mathvariant":5770},[54,14799,62],{"stretchy":79},[44,14801,14802,14804],{},[47,14803,5238],{},[81,14805,107],{},[54,14807,5791],{"separator":61},[44,14809,14810,14812],{},[47,14811,5238],{},[81,14813,158],{},[54,14815,5800],{"stretchy":79},[54,14817,56],{},[41,14819,14820,14822,14896],{},[54,14821,2436],{"fence":61},[64,14823,14824,14860],{"rowspacing":2439,"columnalign":2440,"columnspacing":68},[70,14825,14826,14854],{},[73,14827,14828],{},[76,14829,14830],{"scriptlevel":78,"displaystyle":79},[719,14831,14832,14852],{},[41,14833,14834,14836,14850],{},[54,14835,717],{"fence":61},[703,14837,14838,14848],{},[41,14839,14840,14842,14844],{},[81,14841,107],{},[54,14843,145],{},[708,14845,14846],{},[81,14847,712],{},[81,14849,158],{},[54,14851,741],{"fence":61},[47,14853,52],{"mathvariant":5770},[73,14855,14856],{},[76,14857,14858],{"scriptlevel":78,"displaystyle":79},[81,14859,78],{},[70,14861,14862,14868],{},[73,14863,14864],{},[76,14865,14866],{"scriptlevel":78,"displaystyle":79},[81,14867,78],{},[73,14869,14870],{},[76,14871,14872],{"scriptlevel":78,"displaystyle":79},[719,14873,14874,14894],{},[41,14875,14876,14878,14892],{},[54,14877,717],{"fence":61},[703,14879,14880,14890],{},[41,14881,14882,14884,14886],{},[81,14883,107],{},[54,14885,140],{},[708,14887,14888],{},[81,14889,712],{},[81,14891,158],{},[54,14893,741],{"fence":61},[47,14895,52],{"mathvariant":5770},[54,14897,2471],{"fence":61},[175,14899,14900],{"encoding":177},"\\rm{diag}^n\\{\\lambda_1, \\lambda_2\\} = \\begin{bmatrix}\n\\left(\\frac{1+\\sqrt{5}}{2}\\right)^n & 0 \\\\\n0 & \\left(\\frac{1-\\sqrt{5}}{2}\\right)^n\n\\end{bmatrix}",[19,14902,14904],{"className":14903,"ariaHidden":61},[182],[19,14905,14907,14911],{"className":14906},[186],[19,14908],{"className":14909,"style":14910},[190],"height:3.7086em;vertical-align:-1.6043em;",[19,14912,14914,14946,14949,14989,14992,14995,15035,15038,15041,15044,15047],{"className":14913},[195],[19,14915,14917,14923],{"className":14916},[195],[19,14918,14920],{"className":14919},[195],[19,14921,5824],{"className":14922,"style":5823},[195,5822],[19,14924,14926],{"className":14925},[204],[19,14927,14929],{"className":14928},[208],[19,14930,14932],{"className":14931},[213],[19,14933,14935],{"className":14934,"style":14620},[217],[19,14936,14937,14940],{"style":14623},[19,14938],{"className":14939,"style":226},[225],[19,14941,14943],{"className":14942},[230,231,232,233],[19,14944,52],{"className":14945},[195,5822,233],[19,14947,62],{"className":14948},[277],[19,14950,14952,14955],{"className":14951},[195],[19,14953,5238],{"className":14954},[195,199],[19,14956,14958],{"className":14957},[204],[19,14959,14961,14981],{"className":14960},[208,209],[19,14962,14964,14978],{"className":14963},[213],[19,14965,14967],{"className":14966,"style":450},[217],[19,14968,14969,14972],{"style":5603},[19,14970],{"className":14971,"style":226},[225],[19,14973,14975],{"className":14974},[230,231,232,233],[19,14976,107],{"className":14977},[195,5822,233],[19,14979,241],{"className":14980},[240],[19,14982,14984],{"className":14983},[213],[19,14985,14987],{"className":14986,"style":248},[217],[19,14988],{},[19,14990,5791],{"className":14991},[5871],[19,14993],{"className":14994,"style":971},[254],[19,14996,14998,15001],{"className":14997},[195],[19,14999,5238],{"className":15000},[195,199],[19,15002,15004],{"className":15003},[204],[19,15005,15007,15027],{"className":15006},[208,209],[19,15008,15010,15024],{"className":15009},[213],[19,15011,15013],{"className":15012,"style":450},[217],[19,15014,15015,15018],{"style":5603},[19,15016],{"className":15017,"style":226},[225],[19,15019,15021],{"className":15020},[230,231,232,233],[19,15022,158],{"className":15023},[195,5822,233],[19,15025,241],{"className":15026},[240],[19,15028,15030],{"className":15029},[213],[19,15031,15033],{"className":15032,"style":248},[217],[19,15034],{},[19,15036,5800],{"className":15037},[674],[19,15039],{"className":15040,"style":255},[254],[19,15042,56],{"className":15043},[259],[19,15045],{"className":15046,"style":255},[254],[19,15048,15050,15096,15515],{"className":15049},[273],[19,15051,15053],{"className":15052},[277],[19,15054,15056],{"className":15055},[281,282],[19,15057,15059,15087],{"className":15058},[208,209],[19,15060,15062,15084],{"className":15061},[213],[19,15063,15066],{"className":15064,"style":15065},[217],"height:2.05em;",[19,15067,15069,15073],{"style":15068},"top:-4.05em;",[19,15070],{"className":15071,"style":15072},[225],"height:5.6em;",[19,15074,15076],{"style":15075},"width:0.667em;height:3.600em;",[318,15077,15081],{"xmlns":320,"width":15078,"height":15079,"viewBox":15080},"0.667em","3.600em","0 0 667 3600",[327,15082],{"d":15083},"M403 1759 V84 H666 V0 H319 V1759 v0 v1759 h347 v-84\nH403z M403 1759 V0 H319 V1759 v0 v1759 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only the first row of the matrices on both sides of the equation, we obtain the closed-form formula for the Fibonacci sequence:",[19,19709,19711],{"className":19710,"translate":23},[22],[19,19712,19714,19797],{"className":19713,"translate":23},[27],[19,19715,19717],{"className":19716},[31],[33,19718,19719],{"xmlns":35,"display":36},[38,19720,19721,19795],{},[41,19722,19723,19729,19731,19739],{},[44,19724,19725,19727],{},[47,19726,49],{},[47,19728,52],{},[54,19730,56],{},[703,19732,19733,19735],{},[81,19734,107],{},[708,19736,19737],{},[81,19738,712],{},[41,19740,19741,19743,19767,19769,19793],{},[54,19742,717],{"fence":61},[719,19744,19745,19765],{},[41,19746,19747,19749,19763],{},[54,19748,717],{"fence":61},[703,19750,19751,19761],{},[41,19752,19753,19755,19757],{},[81,19754,107],{},[54,19756,145],{},[708,19758,19759],{},[81,19760,712],{},[81,19762,158],{},[54,19764,741],{"fence":61},[47,19766,52],{},[54,19768,140],{},[719,19770,19771,19791],{},[41,19772,19773,19775,19789],{},[54,19774,717],{"fence":61},[703,19776,19777,19787],{},[41,19778,19779,19781,19783],{},[81,19780,107],{},[54,19782,140],{},[708,19784,19785],{},[81,19786,712],{},[81,19788,158],{},[54,19790,741],{"fence":61},[47,19792,52],{},[54,19794,741],{"fence":61},[175,19796,774],{"encoding":177},[19,19798,19800,19855],{"className":19799,"ariaHidden":61},[182],[19,19801,19803,19806,19846,19849,19852],{"className":19802},[186],[19,19804],{"className":19805,"style":191},[190],[19,19807,19809,19812],{"className":19808},[195],[19,19810,49],{"className":19811,"style":200},[195,199],[19,19813,19815],{"className":19814},[204],[19,19816,19818,19838],{"className":19817},[208,209],[19,19819,19821,19835],{"className":19820},[213],[19,19822,19824],{"className":19823,"style":218},[217],[19,19825,19826,19829],{"style":221},[19,19827],{"className":19828,"style":226},[225],[19,19830,19832],{"className":19831},[230,231,232,233],[19,19833,52],{"className":19834},[195,199,233],[19,19836,241],{"className":19837},[240],[19,19839,19841],{"className":19840},[213],[19,19842,19844],{"className":19843,"style":248},[217],[19,19845],{},[19,19847],{"className":19848,"style":255},[254],[19,19850,56],{"className":19851},[259],[19,19853],{"className":19854,"style":255},[254],[19,19856,19858,19861,19967,19970],{"className":19857},[186],[19,19859],{"className":19860,"style":839},[190],[19,19862,19864,19867,19964],{"className":19863},[195],[19,19865],{"className":19866},[277,675],[19,19868,19870],{"className":19869},[703],[19,19871,19873,19956],{"className":19872},[208,209],[19,19874,19876,19953],{"className":19875},[213],[19,19877,19879,19934,19942],{"className":19878,"style":858},[217],[19,19880,19881,19884],{"style":861},[19,19882],{"className":19883,"style":865},[225],[19,19885,19887],{"className":19886},[195],[19,19888,19890],{"className":19889},[195,872],[19,19891,19893,19926],{"className":19892},[208,209],[19,19894,19896,19923],{"className":19895},[213],[19,19897,19899,19911],{"className":19898,"style":882},[217],[19,19900,19902,19905],{"className":19901,"style":887},[886],[19,19903],{"className":19904,"style":865},[225],[19,19906,19908],{"className":19907,"style":894},[195],[19,19909,712],{"className":19910},[195],[19,19912,19913,19916],{"style":900},[19,19914],{"className":19915,"style":865},[225],[19,19917,19919],{"className":19918,"style":908},[907],[318,19920,19921],{"xmlns":320,"width":911,"height":912,"viewBox":913,"preserveAspectRatio":914},[327,19922],{"d":917},[19,19924,241],{"className":19925},[240],[19,19927,19929],{"className":19928},[213],[19,19930,19932],{"className":19931,"style":927},[217],[19,19933],{},[19,19935,19936,19939],{"style":932},[19,19937],{"className":19938,"style":865},[225],[19,19940],{"className":19941,"style":940},[939],[19,19943,19944,19947],{"style":943},[19,19945],{"className":19946,"style":865},[225],[19,19948,19950],{"className":19949},[195],[19,19951,107],{"className":19952},[195],[19,19954,241],{"className":19955},[240],[19,19957,19959],{"className":19958},[213],[19,19960,19962],{"className":19961,"style":962},[217],[19,19963],{},[19,19965],{"className":19966},[674,675],[19,19968],{"className":19969,"style":971},[254],[19,19971,19973,19979,20138,20141,20144,20147,20306],{"className":19972},[273],[19,19974,19976],{"className":19975,"style":979},[277,978],[19,19977,717],{"className":19978},[281,983],[19,19980,19982,20115],{"className":19981},[273],[19,19983,19985,19991,20109],{"className":19984},[273],[19,19986,19988],{"className":19987,"style":979},[277,978],[19,19989,717],{"className":19990},[281,983],[19,19992,19994,19997,20106],{"className":19993},[195],[19,19995],{"className":19996},[277,675],[19,19998,20000],{"className":19999},[703],[19,20001,20003,20098],{"className":20002},[208,209],[19,20004,20006,20095],{"className":20005},[213],[19,20007,20009,20020,20028],{"className":20008,"style":1014},[217],[19,20010,20011,20014],{"style":1017},[19,20012],{"className":20013,"style":865},[225],[19,20015,20017],{"className":20016},[195],[19,20018,158],{"className":20019},[195],[19,20021,20022,20025],{"style":932},[19,20023],{"className":20024,"style":865},[225],[19,20026],{"className":20027,"style":940},[939],[19,20029,20030,20033],{"style":943},[19,20031],{"className":20032,"style":865},[225],[19,20034,20036,20039,20042,20045,20048],{"className":20035},[195],[19,20037,107],{"className":20038},[195],[19,20040],{"className":20041,"style":487},[254],[19,20043,145],{"className":20044},[468],[19,20046],{"className":20047,"style":487},[254],[19,20049,20051],{"className":20050},[195,872],[19,20052,20054,20087],{"className":20053},[208,209],[19,20055,20057,20084],{"className":20056},[213],[19,20058,20060,20072],{"className":20059,"style":882},[217],[19,20061,20063,20066],{"className":20062,"style":887},[886],[19,20064],{"className":20065,"style":865},[225],[19,20067,20069],{"className":20068,"style":894},[195],[19,20070,712],{"className":20071},[195],[19,20073,20074,20077],{"style":900},[19,20075],{"className":20076,"style":865},[225],[19,20078,20080],{"className":20079,"style":908},[907],[318,20081,20082],{"xmlns":320,"width":911,"height":912,"viewBox":913,"preserveAspectRatio":914},[327,20083],{"d":917},[19,20085,241],{"className":20086},[240],[19,20088,20090],{"className":20089},[213],[19,20091,20093],{"className":20092,"style":927},[217],[19,20094],{},[19,20096,241],{"className":20097},[240],[19,20099,20101],{"className":20100},[213],[19,20102,20104],{"className":20103,"style":1111},[217],[19,20105],{},[19,20107],{"className":20108},[674,675],[19,20110,20112],{"className":20111,"style":979},[674,978],[19,20113,741],{"className":20114},[281,983],[19,20116,20118],{"className":20117},[204],[19,20119,20121],{"className":20120},[208],[19,20122,20124],{"className":20123},[213],[19,20125,20127],{"className":20126,"style":1135},[217],[19,20128,20129,20132],{"style":1138},[19,20130],{"className":20131,"style":226},[225],[19,20133,20135],{"className":20134},[230,231,232,233],[19,20136,52],{"className":20137},[195,199,233],[19,20139],{"className":20140,"style":487},[254],[19,20142,140],{"className":20143},[468],[19,20145],{"className":20146,"style":487},[254],[19,20148,20150,20283],{"className":20149},[273],[19,20151,20153,20159,20277],{"className":20152},[273],[19,20154,20156],{"className":20155,"style":979},[277,978],[19,20157,717],{"className":20158},[281,983],[19,20160,20162,20165,20274],{"className":20161},[195],[19,20163],{"className":20164},[277,675],[19,20166,20168],{"className":20167},[703],[19,20169,20171,20266],{"className":20170},[208,209],[19,20172,20174,20263],{"className":20173},[213],[19,20175,20177,20188,20196],{"className":20176,"style":1014},[217],[19,20178,20179,20182],{"style":1017},[19,20180],{"className":20181,"style":865},[225],[19,20183,20185],{"className":20184},[195],[19,20186,158],{"className":20187},[195],[19,20189,20190,20193],{"style":932},[19,20191],{"className":20192,"style":865},[225],[19,20194],{"className":20195,"style":940},[939],[19,20197,20198,20201],{"style":943},[19,20199],{"className":20200,"style":865},[225],[19,20202,20204,20207,20210,20213,20216],{"className":20203},[195],[19,20205,107],{"className":20206},[195],[19,20208],{"className":20209,"style":487},[254],[19,20211,140],{"className":20212},[468],[19,20214],{"className":20215,"style":487},[254],[19,20217,20219],{"className":20218},[195,872],[19,20220,20222,20255],{"className":20221},[208,209],[19,20223,20225,20252],{"className":20224},[213],[19,20226,20228,20240],{"className":20227,"style":882},[217],[19,20229,20231,20234],{"className":20230,"style":887},[886],[19,20232],{"className":20233,"style":865},[225],[19,20235,20237],{"className":20236,"style":894},[195],[19,20238,712],{"className":20239},[195],[19,20241,20242,20245],{"style":900},[19,20243],{"className":20244,"style":865},[225],[19,20246,20248],{"className":20247,"style":908},[907],[318,20249,20250],{"xmlns":320,"width":911,"height":912,"viewBox":913,"preserveAspectRatio":914},[327,20251],{"d":917},[19,20253,241],{"className":20254},[240],[19,20256,20258],{"className":20257},[213],[19,20259,20261],{"className":20260,"style":927},[217],[19,20262],{},[19,20264,241],{"className":20265},[240],[19,20267,20269],{"className":20268},[213],[19,20270,20272],{"className":20271,"style":1111},[217],[19,20273],{},[19,20275],{"className":20276},[674,675],[19,20278,20280],{"className":20279,"style":979},[674,978],[19,20281,741],{"className":20282},[281,983],[19,20284,20286],{"className":20285},[204],[19,20287,20289],{"className":20288},[208],[19,20290,20292],{"className":20291},[213],[19,20293,20295],{"className":20294,"style":1135},[217],[19,20296,20297,20300],{"style":1138},[19,20298],{"className":20299,"style":226},[225],[19,20301,20303],{"className":20302},[230,231,232,233],[19,20304,52],{"className":20305},[195,199,233],[19,20307,20309],{"className":20308,"style":979},[674,978],[19,20310,741],{"className":20311},[281,983],[10,20313,20314,20315,5624,20366,20417],{},"Substituting ",[19,20316,20318,20336],{"className":20317,"translate":23},[27],[19,20319,20321],{"className":20320},[31],[33,20322,20323],{"xmlns":35},[38,20324,20325,20333],{},[41,20326,20327,20329,20331],{},[47,20328,52],{},[54,20330,56],{},[81,20332,78],{},[175,20334,20335],{"encoding":177},"n=0",[19,20337,20339,20357],{"className":20338,"ariaHidden":61},[182],[19,20340,20342,20345,20348,20351,20354],{"className":20341},[186],[19,20343],{"className":20344,"style":5104},[190],[19,20346,52],{"className":20347},[195,199],[19,20349],{"className":20350,"style":255},[254],[19,20352,56],{"className":20353},[259],[19,20355],{"className":20356,"style":255},[254],[19,20358,20360,20363],{"className":20359},[186],[19,20361],{"className":20362,"style":1382},[190],[19,20364,78],{"className":20365},[195],[19,20367,20369,20387],{"className":20368,"translate":23},[27],[19,20370,20372],{"className":20371},[31],[33,20373,20374],{"xmlns":35},[38,20375,20376,20384],{},[41,20377,20378,20380,20382],{},[47,20379,52],{},[54,20381,56],{},[81,20383,107],{},[175,20385,20386],{"encoding":177},"n=1",[19,20388,20390,20408],{"className":20389,"ariaHidden":61},[182],[19,20391,20393,20396,20399,20402,20405],{"className":20392},[186],[19,20394],{"className":20395,"style":5104},[190],[19,20397,52],{"className":20398},[195,199],[19,20400],{"className":20401,"style":255},[254],[19,20403,56],{"className":20404},[259],[19,20406],{"className":20407,"style":255},[254],[19,20409,20411,20414],{"className":20410},[186],[19,20412],{"className":20413,"style":1382},[190],[19,20415,107],{"className":20416},[195]," to verify, we find that they both satisfy the equation.",[20419,20420],"hr",{},[1326,20422,20424],{"id":20423},"difference-equation-method","Difference Equation Method",[10,20426,20427,20428,20508,20509,20728],{},"Defining the difference of a sequence ",[19,20429,20431,20453],{"className":20430,"translate":23},[27],[19,20432,20434],{"className":20433},[31],[33,20435,20436],{"xmlns":35},[38,20437,20438,20450],{},[41,20439,20440,20442,20448],{},[54,20441,62],{"stretchy":79},[44,20443,20444,20446],{},[47,20445,5777],{},[47,20447,52],{},[54,20449,5800],{"stretchy":79},[175,20451,20452],{"encoding":177},"\\{a_n\\}",[19,20454,20456],{"className":20455,"ariaHidden":61},[182],[19,20457,20459,20462,20465,20505],{"className":20458},[186],[19,20460],{"className":20461,"style":5312},[190],[19,20463,62],{"className":20464},[277],[19,20466,20468,20471],{"className":20467},[195],[19,20469,5777],{"className":20470},[195,199],[19,20472,20474],{"className":20473},[204],[19,20475,20477,20497],{"className":20476},[208,209],[19,20478,20480,20494],{"className":20479},[213],[19,20481,20483],{"className":20482,"style":218},[217],[19,20484,20485,20488],{"style":5603},[19,20486],{"className":20487,"style":226},[225],[19,20489,20491],{"className":20490},[230,231,232,233],[19,20492,52],{"className":20493},[195,199,233],[19,20495,241],{"className":20496},[240],[19,20498,20500],{"className":20499},[213],[19,20501,20503],{"className":20502,"style":248},[217],[19,20504],{},[19,20506,5800],{"className":20507},[674]," as ",[19,20510,20512,20555],{"className":20511,"translate":23},[27],[19,20513,20515],{"className":20514},[31],[33,20516,20517],{"xmlns":35},[38,20518,20519,20552],{},[41,20520,20521,20524,20530,20532,20544,20546],{},[47,20522,20523],{"mathvariant":5770},"Δ",[44,20525,20526,20528],{},[47,20527,5777],{},[47,20529,52],{},[54,20531,56],{},[44,20533,20534,20536],{},[47,20535,5777],{},[41,20537,20538,20540,20542],{},[47,20539,52],{},[54,20541,145],{},[81,20543,107],{},[54,20545,140],{},[44,20547,20548,20550],{},[47,20549,5777],{},[47,20551,52],{},[175,20553,20554],{"encoding":177},"\\Delta a_n = a_{n+1} - a_n",[19,20556,20558,20616,20681],{"className":20557,"ariaHidden":61},[182],[19,20559,20561,20564,20567,20607,20610,20613],{"className":20560},[186],[19,20562],{"className":20563,"style":191},[190],[19,20565,20523],{"className":20566},[195],[19,20568,20570,20573],{"className":20569},[195],[19,20571,5777],{"className":20572},[195,199],[19,20574,20576],{"className":20575},[204],[19,20577,20579,20599],{"className":20578},[208,209],[19,20580,20582,20596],{"className":20581},[213],[19,20583,20585],{"className":20584,"style":218},[217],[19,20586,20587,20590],{"style":5603},[19,20588],{"className":20589,"style":226},[225],[19,20591,20593],{"className":20592},[230,231,232,233],[19,20594,52],{"className":20595},[195,199,233],[19,20597,241],{"className":20598},[240],[19,20600,20602],{"className":20601},[213],[19,20603,20605],{"className":20604,"style":248},[217],[19,20606],{},[19,20608],{"className":20609,"style":255},[254],[19,20611,56],{"className":20612},[259],[19,20614],{"className":20615,"style":255},[254],[19,20617,20619,20623,20672,20675,20678],{"className":20618},[186],[19,20620],{"className":20621,"style":20622},[190],"height:0.7917em;vertical-align:-0.2083em;",[19,20624,20626,20629],{"className":20625},[195],[19,20627,5777],{"className":20628},[195,199],[19,20630,20632],{"className":20631},[204],[19,20633,20635,20664],{"className":20634},[208,209],[19,20636,20638,20661],{"className":20637},[213],[19,20639,20641],{"className":20640,"style":450},[217],[19,20642,20643,20646],{"style":5603},[19,20644],{"className":20645,"style":226},[225],[19,20647,20649],{"className":20648},[230,231,232,233],[19,20650,20652,20655,20658],{"className":20651},[195,233],[19,20653,52],{"className":20654},[195,199,233],[19,20656,145],{"className":20657},[468,233],[19,20659,107],{"className":20660},[195,233],[19,20662,241],{"className":20663},[240],[19,20665,20667],{"className":20666},[213],[19,20668,20670],{"className":20669,"style":481},[217],[19,20671],{},[19,20673],{"className":20674,"style":487},[254],[19,20676,140],{"className":20677},[468],[19,20679],{"className":20680,"style":487},[254],[19,20682,20684,20688],{"className":20683},[186],[19,20685],{"className":20686,"style":20687},[190],"height:0.5806em;vertical-align:-0.15em;",[19,20689,20691,20694],{"className":20690},[195],[19,20692,5777],{"className":20693},[195,199],[19,20695,20697],{"className":20696},[204],[19,20698,20700,20720],{"className":20699},[208,209],[19,20701,20703,20717],{"className":20702},[213],[19,20704,20706],{"className":20705,"style":218},[217],[19,20707,20708,20711],{"style":5603},[19,20709],{"className":20710,"style":226},[225],[19,20712,20714],{"className":20713},[230,231,232,233],[19,20715,52],{"className":20716},[195,199,233],[19,20718,241],{"className":20719},[240],[19,20721,20723],{"className":20722},[213],[19,20724,20726],{"className":20725,"style":248},[217],[19,20727],{}," (using backward difference here), we can define the second-order difference as:",[19,20730,20732],{"className":20731,"translate":23},[22],[19,20733,20735,20823],{"className":20734,"translate":23},[27],[19,20736,20738],{"className":20737},[31],[33,20739,20740],{"xmlns":35,"display":36},[38,20741,20742,20820],{},[41,20743,20744,20750,20756,20758,20760,20772,20774,20776,20782,20784,20796,20798,20800,20812,20814],{},[719,20745,20746,20748],{},[47,20747,20523],{"mathvariant":5770},[81,20749,158],{},[44,20751,20752,20754],{},[47,20753,5777],{},[47,20755,52],{},[54,20757,56],{},[47,20759,20523],{"mathvariant":5770},[44,20761,20762,20764],{},[47,20763,5777],{},[41,20765,20766,20768,20770],{},[47,20767,52],{},[54,20769,145],{},[81,20771,107],{},[54,20773,140],{},[47,20775,20523],{"mathvariant":5770},[44,20777,20778,20780],{},[47,20779,5777],{},[47,20781,52],{},[54,20783,56],{},[44,20785,20786,20788],{},[47,20787,5777],{},[41,20789,20790,20792,20794],{},[47,20791,52],{},[54,20793,145],{},[81,20795,158],{},[54,20797,140],{},[81,20799,158],{},[44,20801,20802,20804],{},[47,20803,5777],{},[41,20805,20806,20808,20810],{},[47,20807,52],{},[54,20809,145],{},[81,20811,107],{},[54,20813,145],{},[44,20815,20816,20818],{},[47,20817,5777],{},[47,20819,52],{},[175,20821,20822],{"encoding":177},"\\Delta^2 a_n = \\Delta a_{n+1} - \\Delta a_n = a_{n+2} - 2a_{n+1} + a_n",[19,20824,20826,20911,20978,21036,21100,21168],{"className":20825,"ariaHidden":61},[182],[19,20827,20829,20833,20862,20902,20905,20908],{"className":20828},[186],[19,20830],{"className":20831,"style":20832},[190],"height:1.0141em;vertical-align:-0.15em;",[19,20834,20836,20839],{"className":20835},[195],[19,20837,20523],{"className":20838},[195],[19,20840,20842],{"className":20841},[204],[19,20843,20845],{"className":20844},[208],[19,20846,20848],{"className":20847},[213],[19,20849,20851],{"className":20850,"style":7748},[217],[19,20852,20853,20856],{"style":6430},[19,20854],{"className":20855,"style":226},[225],[19,20857,20859],{"className":20858},[230,231,232,233],[19,20860,158],{"className":20861},[195,233],[19,20863,20865,20868],{"className":20864},[195],[19,20866,5777],{"className":20867},[195,199],[19,20869,20871],{"className":20870},[204],[19,20872,20874,20894],{"className":20873},[208,209],[19,20875,20877,20891],{"className":20876},[213],[19,20878,20880],{"className":20879,"style":218},[217],[19,20881,20882,20885],{"style":5603},[19,20883],{"className":20884,"style":226},[225],[19,20886,20888],{"className":20887},[230,231,232,233],[19,20889,52],{"className":20890},[195,199,233],[19,20892,241],{"className":20893},[240],[19,20895,20897],{"className":20896},[213],[19,20898,20900],{"className":20899,"style":248},[217],[19,20901],{},[19,20903],{"className":20904,"style":255},[254],[19,20906,56],{"className":20907},[259],[19,20909],{"className":20910,"style":255},[254],[19,20912,20914,20917,20920,20969,20972,20975],{"className":20913},[186],[19,20915],{"className":20916,"style":1451},[190],[19,20918,20523],{"className":20919},[195],[19,20921,20923,20926],{"className":20922},[195],[19,20924,5777],{"className":20925},[195,199],[19,20927,20929],{"className":20928},[204],[19,20930,20932,20961],{"className":20931},[208,209],[19,20933,20935,20958],{"className":20934},[213],[19,20936,20938],{"className":20937,"style":450},[217],[19,20939,20940,20943],{"style":5603},[19,20941],{"className":20942,"style":226},[225],[19,20944,20946],{"className":20945},[230,231,232,233],[19,20947,20949,20952,20955],{"className":20948},[195,233],[19,20950,52],{"className":20951},[195,199,233],[19,20953,145],{"className":20954},[468,233],[19,20956,107],{"className":20957},[195,233],[19,20959,241],{"className":20960},[240],[19,20962,20964],{"className":20963},[213],[19,20965,20967],{"className":20966,"style":481},[217],[19,20968],{},[19,20970],{"className":20971,"style":487},[254],[19,20973,140],{"className":20974},[468],[19,20976],{"className":20977,"style":487},[254],[19,20979,20981,20984,20987,21027,21030,21033],{"className":20980},[186],[19,20982],{"className":20983,"style":191},[190],[19,20985,20523],{"className":20986},[195],[19,20988,20990,20993],{"className":20989},[195],[19,20991,5777],{"className":20992},[195,199],[19,20994,20996],{"className":20995},[204],[19,20997,20999,21019],{"className":20998},[208,209],[19,21000,21002,21016],{"className":21001},[213],[19,21003,21005],{"className":21004,"style":218},[217],[19,21006,21007,21010],{"style":5603},[19,21008],{"className":21009,"style":226},[225],[19,21011,21013],{"className":21012},[230,231,232,233],[19,21014,52],{"className":21015},[195,199,233],[19,21017,241],{"className":21018},[240],[19,21020,21022],{"className":21021},[213],[19,21023,21025],{"className":21024,"style":248},[217],[19,21026],{},[19,21028],{"className":21029,"style":255},[254],[19,21031,56],{"className":21032},[259],[19,21034],{"className":21035,"style":255},[254],[19,21037,21039,21042,21091,21094,21097],{"className":21038},[186],[19,21040],{"className":21041,"style":20622},[190],[19,21043,21045,21048],{"className":21044},[195],[19,21046,5777],{"className":21047},[195,199],[19,21049,21051],{"className":21050},[204],[19,21052,21054,21083],{"className":21053},[208,209],[19,21055,21057,21080],{"className":21056},[213],[19,21058,21060],{"className":21059,"style":450},[217],[19,21061,21062,21065],{"style":5603},[19,21063],{"className":21064,"style":226},[225],[19,21066,21068],{"className":21067},[230,231,232,233],[19,21069,21071,21074,21077],{"className":21070},[195,233],[19,21072,52],{"className":21073},[195,199,233],[19,21075,145],{"className":21076},[468,233],[19,21078,158],{"className":21079},[195,233],[19,21081,241],{"className":21082},[240],[19,21084,21086],{"className":21085},[213],[19,21087,21089],{"className":21088,"style":481},[217],[19,21090],{},[19,21092],{"className":21093,"style":487},[254],[19,21095,140],{"className":21096},[468],[19,21098],{"className":21099,"style":487},[254],[19,21101,21103,21107,21110,21159,21162,21165],{"className":21102},[186],[19,21104],{"className":21105,"style":21106},[190],"height:0.8528em;vertical-align:-0.2083em;",[19,21108,158],{"className":21109},[195],[19,21111,21113,21116],{"className":21112},[195],[19,21114,5777],{"className":21115},[195,199],[19,21117,21119],{"className":21118},[204],[19,21120,21122,21151],{"className":21121},[208,209],[19,21123,21125,21148],{"className":21124},[213],[19,21126,21128],{"className":21127,"style":450},[217],[19,21129,21130,21133],{"style":5603},[19,21131],{"className":21132,"style":226},[225],[19,21134,21136],{"className":21135},[230,231,232,233],[19,21137,21139,21142,21145],{"className":21138},[195,233],[19,21140,52],{"className":21141},[195,199,233],[19,21143,145],{"className":21144},[468,233],[19,21146,107],{"className":21147},[195,233],[19,21149,241],{"className":21150},[240],[19,21152,21154],{"className":21153},[213],[19,21155,21157],{"className":21156,"style":481},[217],[19,21158],{},[19,21160],{"className":21161,"style":487},[254],[19,21163,145],{"className":21164},[468],[19,21166],{"className":21167,"style":487},[254],[19,21169,21171,21174],{"className":21170},[186],[19,21172],{"className":21173,"style":20687},[190],[19,21175,21177,21180],{"className":21176},[195],[19,21178,5777],{"className":21179},[195,199],[19,21181,21183],{"className":21182},[204],[19,21184,21186,21206],{"className":21185},[208,209],[19,21187,21189,21203],{"className":21188},[213],[19,21190,21192],{"className":21191,"style":218},[217],[19,21193,21194,21197],{"style":5603},[19,21195],{"className":21196,"style":226},[225],[19,21198,21200],{"className":21199},[230,231,232,233],[19,21201,52],{"className":21202},[195,199,233],[19,21204,241],{"className":21205},[240],[19,21207,21209],{"className":21208},[213],[19,21210,21212],{"className":21211,"style":248},[217],[19,21213],{},[10,21215,21216,21217,21246],{},"Further, we can define the ",[19,21218,21220,21234],{"className":21219,"translate":23},[27],[19,21221,21223],{"className":21222},[31],[33,21224,21225],{"xmlns":35},[38,21226,21227,21232],{},[41,21228,21229],{},[47,21230,21231],{},"m",[175,21233,21231],{"encoding":177},[19,21235,21237],{"className":21236,"ariaHidden":61},[182],[19,21238,21240,21243],{"className":21239},[186],[19,21241],{"className":21242,"style":5104},[190],[19,21244,21231],{"className":21245},[195,199],"-th order difference:",[19,21248,21250],{"className":21249,"translate":23},[22],[19,21251,21253,21379],{"className":21252,"translate":23},[27],[19,21254,21256],{"className":21255},[31],[33,21257,21258],{"xmlns":35,"display":36},[38,21259,21260,21376],{},[41,21261,21262,21268,21274,21276,21288,21300,21302,21314,21320,21322,21338,21340,21342,21344,21350,21360],{},[719,21263,21264,21266],{},[47,21265,20523],{"mathvariant":5770},[47,21267,21231],{},[44,21269,21270,21272],{},[47,21271,5777],{},[47,21273,52],{},[54,21275,56],{},[719,21277,21278,21280],{},[47,21279,20523],{"mathvariant":5770},[41,21281,21282,21284,21286],{},[47,21283,21231],{},[54,21285,140],{},[81,21287,107],{},[44,21289,21290,21292],{},[47,21291,5777],{},[41,21293,21294,21296,21298],{},[47,21295,52],{},[54,21297,145],{},[81,21299,107],{},[54,21301,140],{},[719,21303,21304,21306],{},[47,21305,20523],{"mathvariant":5770},[41,21307,21308,21310,21312],{},[47,21309,21231],{},[54,21311,140],{},[81,21313,107],{},[44,21315,21316,21318],{},[47,21317,5777],{},[47,21319,52],{},[54,21321,56],{},[21323,21324,21325,21328,21336],"munderover",{},[54,21326,21327],{},"∑",[41,21329,21330,21332,21334],{},[47,21331,5774],{},[54,21333,56],{},[81,21335,78],{},[47,21337,21231],{},[54,21339,717],{"stretchy":79},[54,21341,140],{},[81,21343,107],{},[719,21345,21346,21348],{},[54,21347,741],{"stretchy":79},[47,21349,5774],{},[21351,21352,21353,21356,21358],"msubsup",{},[47,21354,21355],{},"C",[47,21357,21231],{},[47,21359,5774],{},[44,21361,21362,21364],{},[47,21363,5777],{},[41,21365,21366,21368,21370,21372,21374],{},[47,21367,52],{},[54,21369,145],{},[47,21371,21231],{},[54,21373,140],{},[47,21375,5774],{},[175,21377,21378],{"encoding":177},"\\Delta^m a_n = \\Delta^{m-1} a_{n+1} - \\Delta^{m-1} a_n = \\sum_{i=0}^{m} (-1)^i C_m^i a_{n+m-i}",[19,21380,21382,21467,21570,21663],{"className":21381,"ariaHidden":61},[182],[19,21383,21385,21389,21418,21458,21461,21464],{"className":21384},[186],[19,21386],{"className":21387,"style":21388},[190],"height:0.8644em;vertical-align:-0.15em;",[19,21390,21392,21395],{"className":21391},[195],[19,21393,20523],{"className":21394},[195],[19,21396,21398],{"className":21397},[204],[19,21399,21401],{"className":21400},[208],[19,21402,21404],{"className":21403},[213],[19,21405,21407],{"className":21406,"style":6427},[217],[19,21408,21409,21412],{"style":6430},[19,21410],{"className":21411,"style":226},[225],[19,21413,21415],{"className":21414},[230,231,232,233],[19,21416,21231],{"className":21417},[195,199,233],[19,21419,21421,21424],{"className":21420},[195],[19,21422,5777],{"className":21423},[195,199],[19,21425,21427],{"className":21426},[204],[19,21428,21430,21450],{"className":21429},[208,209],[19,21431,21433,21447],{"className":21432},[213],[19,21434,21436],{"className":21435,"style":218},[217],[19,21437,21438,21441],{"style":5603},[19,21439],{"className":21440,"style":226},[225],[19,21442,21444],{"className":21443},[230,231,232,233],[19,21445,52],{"className":21446},[195,199,233],[19,21448,241],{"className":21449},[240],[19,21451,21453],{"className":21452},[213],[19,21454,21456],{"className":21455,"style":248},[217],[19,21457],{},[19,21459],{"className":21460,"style":255},[254],[19,21462,56],{"className":21463},[259],[19,21465],{"className":21466,"style":255},[254],[19,21468,21470,21474,21512,21561,21564,21567],{"className":21469},[186],[19,21471],{"className":21472,"style":21473},[190],"height:1.0724em;vertical-align:-0.2083em;",[19,21475,21477,21480],{"className":21476},[195],[19,21478,20523],{"className":21479},[195],[19,21481,21483],{"className":21482},[204],[19,21484,21486],{"className":21485},[208],[19,21487,21489],{"className":21488},[213],[19,21490,21492],{"className":21491,"style":7748},[217],[19,21493,21494,21497],{"style":6430},[19,21495],{"className":21496,"style":226},[225],[19,21498,21500],{"className":21499},[230,231,232,233],[19,21501,21503,21506,21509],{"className":21502},[195,233],[19,21504,21231],{"className":21505},[195,199,233],[19,21507,140],{"className":21508},[468,233],[19,21510,107],{"className":21511},[195,233],[19,21513,21515,21518],{"className":21514},[195],[19,21516,5777],{"className":21517},[195,199],[19,21519,21521],{"className":21520},[204],[19,21522,21524,21553],{"className":21523},[208,209],[19,21525,21527,21550],{"className":21526},[213],[19,21528,21530],{"className":21529,"style":450},[217],[19,21531,21532,21535],{"style":5603},[19,21533],{"className":21534,"style":226},[225],[19,21536,21538],{"className":21537},[230,231,232,233],[19,21539,21541,21544,21547],{"className":21540},[195,233],[19,21542,52],{"className":21543},[195,199,233],[19,21545,145],{"className":21546},[468,233],[19,21548,107],{"className":21549},[195,233],[19,21551,241],{"className":21552},[240],[19,21554,21556],{"className":21555},[213],[19,21557,21559],{"className":21558,"style":481},[217],[19,21560],{},[19,21562],{"className":21563,"style":487},[254],[19,21565,140],{"className":21566},[468],[19,21568],{"className":21569,"style":487},[254],[19,21571,21573,21576,21614,21654,21657,21660],{"className":21572},[186],[19,21574],{"className":21575,"style":20832},[190],[19,21577,21579,21582],{"className":21578},[195],[19,21580,20523],{"className":21581},[195],[19,21583,21585],{"className":21584},[204],[19,21586,21588],{"className":21587},[208],[19,21589,21591],{"className":21590},[213],[19,21592,21594],{"className":21593,"style":7748},[217],[19,21595,21596,21599],{"style":6430},[19,21597],{"className":21598,"style":226},[225],[19,21600,21602],{"className":21601},[230,231,232,233],[19,21603,21605,21608,21611],{"className":21604},[195,233],[19,21606,21231],{"className":21607},[195,199,233],[19,21609,140],{"className":21610},[468,233],[19,21612,107],{"className":21613},[195,233],[19,21615,21617,21620],{"className":21616},[195],[19,21618,5777],{"className":21619},[195,199],[19,21621,21623],{"className":21622},[204],[19,21624,21626,21646],{"className":21625},[208,209],[19,21627,21629,21643],{"className":21628},[213],[19,21630,21632],{"className":21631,"style":218},[217],[19,21633,21634,21637],{"style":5603},[19,21635],{"className":21636,"style":226},[225],[19,21638,21640],{"className":21639},[230,231,232,233],[19,21641,52],{"className":21642},[195,199,233],[19,21644,241],{"className":21645},[240],[19,21647,21649],{"className":21648},[213],[19,21650,21652],{"className":21651,"style":248},[217],[19,21653],{},[19,21655],{"className":21656,"style":255},[254],[19,21658,56],{"className":21659},[259],[19,21661],{"className":21662,"style":255},[254],[19,21664,21666,21670,21747,21750,21753,21756,21786,21840],{"className":21665},[186],[19,21667],{"className":21668,"style":21669},[190],"height:2.9291em;vertical-align:-1.2777em;",[19,21671,21675],{"className":21672},[21673,21674],"mop","op-limits",[19,21676,21678,217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F_{n-2}",[19,22351,22353,22408,22472],{"className":22352,"ariaHidden":61},[182],[19,22354,22356,22359,22399,22402,22405],{"className":22355},[186],[19,22357],{"className":22358,"style":191},[190],[19,22360,22362,22365],{"className":22361},[195],[19,22363,49],{"className":22364,"style":200},[195,199],[19,22366,22368],{"className":22367},[204],[19,22369,22371,22391],{"className":22370},[208,209],[19,22372,22374,22388],{"className":22373},[213],[19,22375,22377],{"className":22376,"style":218},[217],[19,22378,22379,22382],{"style":221},[19,22380],{"className":22381,"style":226},[225],[19,22383,22385],{"className":22384},[230,231,232,233],[19,22386,52],{"className":22387},[195,199,233],[19,22389,241],{"className":22390},[240],[19,22392,22394],{"className":22393},[213],[19,22395,22397],{"className":22396,"style":248},[217],[19,22398],{},[19,22400],{"className":22401,"style":255},[254],[19,22403,56],{"className":22404},[259],[19,22406],{"className":22407,"style":255},[254],[19,22409,22411,22414,22463,22466,22469],{"className":22410},[186],[19,22412],{"className":22413,"style":1451},[190],[19,22415,22417,22420],{"className":22416},[195],[19,22418,49],{"className":22419,"style":200},[195,199],[19,22421,22423],{"className":22422},[204],[19,22424,22426,22455],{"className":22425},[208,209],[19,22427,22429,22452],{"className":22428},[213],[19,22430,22432],{"className":22431,"style":450},[217],[19,22433,22434,22437],{"style":221},[19,22435],{"className":22436,"style":226},[225],[19,22438,22440],{"className":22439},[230,231,232,233],[19,22441,22443,22446,22449],{"className":22442},[195,233],[19,22444,52],{"className":22445},[195,199,233],[19,22447,140],{"className":22448},[468,233],[19,22450,107],{"className":22451},[195,233],[19,22453,241],{"className":22454},[240],[19,22456,22458],{"className":22457},[213],[19,22459,22461],{"className":22460,"style":481},[217],[19,22462],{},[19,22464],{"className":22465,"style":487},[254],[19,22467,145],{"className":22468},[468],[19,22470],{"className":22471,"style":487},[254],[19,22473,22475,22478],{"className":22474},[186],[19,22476],{"className":22477,"style":1451},[190],[19,22479,22481,22484],{"className":22480},[195],[19,22482,49],{"className":22483,"style":200},[195,199],[19,22485,22487],{"className":22486},[204],[19,22488,22490,22519],{"className":22489},[208,209],[19,22491,22493,22516],{"className":22492},[213],[19,22494,22496],{"className":22495,"style":450},[217],[19,22497,22498,22501],{"style":221},[19,22499],{"className":22500,"style":226},[225],[19,22502,22504],{"className":22503},[230,231,232,233],[19,22505,22507,22510,22513],{"className":22506},[195,233],[19,22508,52],{"className":22509},[195,199,233],[19,22511,140],{"className":22512},[468,233],[19,22514,158],{"className":22515},[195,233],[19,22517,241],{"className":22518},[240],[19,22520,22522],{"className":22521},[213],[19,22523,22525],{"className":22524,"style":481},[217],[19,22526],{},", has a second-order difference that is constant:",[19,22529,22531],{"className":22530,"translate":23},[22],[19,22532,22534,22678],{"className":22533,"translate":23},[27],[19,22535,22537],{"className":22536},[31],[33,22538,22539],{"xmlns":35,"display":36},[38,22540,22541,22675],{},[41,22542,22543,22549,22555,22557,22569,22571,22573,22585,22587,22593,22595,22607,22609,22621,22623,22625,22637,22639,22645,22647,22659,22661,22667,22669],{},[719,22544,22545,22547],{},[47,22546,20523],{"mathvariant":5770},[81,22548,158],{},[44,22550,22551,22553],{},[47,22552,49],{},[47,22554,52],{},[54,22556,56],{},[44,22558,22559,22561],{},[47,22560,49],{},[41,22562,22563,22565,22567],{},[47,22564,52],{},[54,22566,145],{},[81,22568,158],{},[54,22570,140],{},[81,22572,158],{},[44,22574,22575,22577],{},[47,22576,49],{},[41,22578,22579,22581,22583],{},[47,22580,52],{},[54,22582,145],{},[81,22584,107],{},[54,22586,145],{},[44,22588,22589,22591],{},[47,22590,49],{},[47,22592,52],{},[54,22594,56],{},[44,22596,22597,22599],{},[47,22598,49],{},[41,22600,22601,22603,22605],{},[47,22602,52],{},[54,22604,145],{},[81,22606,107],{},[54,22608,145],{},[44,22610,22611,22613],{},[47,22612,49],{},[41,22614,22615,22617,22619],{},[47,22616,52],{},[54,22618,145],{},[81,22620,107],{},[54,22622,140],{},[81,22624,158],{},[44,22626,22627,22629],{},[47,22628,49],{},[41,22630,22631,22633,22635],{},[47,22632,52],{},[54,22634,145],{},[81,22636,107],{},[54,22638,145],{},[44,22640,22641,22643],{},[47,22642,49],{},[47,22644,52],{},[54,22646,56],{},[44,22648,22649,22651],{},[47,22650,49],{},[41,22652,22653,22655,22657],{},[47,22654,52],{},[54,22656,145],{},[81,22658,107],{},[54,22660,140],{},[44,22662,22663,22665],{},[47,22664,49],{},[47,22666,52],{},[54,22668,56],{},[44,22670,22671,22673],{},[47,22672,49],{},[47,22674,52],{},[175,22676,22677],{"encoding":177},"\\Delta^2 F_n = F_{n+2} - 2F_{n+1} + F_n = F_{n+1} + F_{n+1} - 2F_{n+1} + F_n = F_{n+1} - F_n = F_n",[19,22679,22681,22765,22829,22896,22951,23015,23079,23146,23201,23265,23320],{"className":22680,"ariaHidden":61},[182],[19,22682,22684,22687,22716,22756,22759,22762],{"className":22683},[186],[19,22685],{"className":22686,"style":20832},[190],[19,22688,22690,22693],{"className":22689},[195],[19,22691,20523],{"className":22692},[195],[19,22694,22696],{"className":22695},[204],[19,22697,22699],{"className":22698},[208],[19,22700,22702],{"className":22701},[213],[19,22703,22705],{"className":22704,"style":7748},[217],[19,22706,22707,22710],{"style":6430},[19,22708],{"className":22709,"style":226},[225],[19,22711,22713],{"className":22712},[230,231,232,233],[19,22714,158],{"className":22715},[195,233],[19,22717,22719,22722],{"className":22718},[195],[19,22720,49],{"className":22721,"style":200},[195,199],[19,22723,22725],{"className":22724},[204],[19,22726,22728,22748],{"className":22727},[208,209],[19,22729,22731,22745],{"className":22730},[213],[19,22732,22734],{"className":22733,"style":218},[217],[19,22735,22736,22739],{"style":221},[19,22737],{"className":22738,"style":226},[225],[19,22740,22742],{"className":22741},[230,231,232,233],[19,22743,52],{"className":22744},[195,199,233],[19,22746,241],{"className":22747},[240],[19,22749,22751],{"className":22750},[213],[19,22752,22754],{"className":22753,"style":248},[217],[19,22755],{},[19,22757],{"className":22758,"style":255},[254],[19,22760,56],{"className":22761},[259],[19,22763],{"className":22764,"style":255},[254],[19,22766,22768,22771,22820,22823,22826],{"className":22767},[186],[19,22769],{"className":22770,"style":1451},[190],[19,22772,22774,22777],{"className":22773},[195],[19,22775,49],{"className":22776,"style":200},[195,199],[19,22778,22780],{"className":22779},[204],[19,22781,22783,22812],{"className":22782},[208,209],[19,22784,22786,22809],{"className":22785},[213],[19,22787,22789],{"className":22788,"style":450},[217],[19,22790,22791,22794],{"style":221},[19,22792],{"className":22793,"style":226},[225],[19,22795,22797],{"className":22796},[230,231,232,233],[19,22798,22800,22803,22806],{"className":22799},[195,233],[19,22801,52],{"className":22802},[195,199,233],[19,22804,145],{"className":22805},[468,233],[19,22807,158],{"className":22808},[195,233],[19,22810,241],{"className":22811},[240],[19,22813,22815],{"className":22814},[213],[19,22816,22818],{"className":22817,"style":481},[217],[19,22819],{},[19,22821],{"className":22822,"style":487},[254],[19,22824,140],{"className":22825},[468],[19,22827],{"className":22828,"style":487},[254],[19,22830,22832,22835,22838,22887,22890,22893],{"className":22831},[186],[19,22833],{"className":22834,"style":1451},[190],[19,22836,158],{"className":22837},[195],[19,22839,22841,22844],{"className":22840},[195],[19,22842,49],{"className":22843,"style":200},[195,199],[19,22845,22847],{"className":22846},[204],[19,22848,22850,22879],{"className":22849},[208,209],[19,22851,22853,22876],{"className":22852},[213],[19,22854,22856],{"className":22855,"style":450},[217],[19,22857,22858,22861],{"style":221},[19,22859],{"className":22860,"style":226},[225],[19,22862,22864],{"className":22863},[230,231,232,233],[19,22865,22867,22870,22873],{"className":22866},[195,233],[19,22868,52],{"className":22869},[195,199,233],[19,22871,145],{"className":22872},[468,233],[19,22874,107],{"className":22875},[195,233],[19,22877,241],{"className":22878},[240],[19,22880,22882],{"className":22881},[213],[19,22883,22885],{"className":22884,"style":481},[217],[19,22886],{},[19,22888],{"className":22889,"style":487},[254],[19,22891,145],{"className":22892},[468],[19,22894],{"className":22895,"style":487},[254],[19,22897,22899,22902,22942,22945,22948],{"className":22898},[186],[19,22900],{"className":22901,"style":191},[190],[19,22903,22905,22908],{"className":22904},[195],[19,22906,49],{"className":22907,"style":200},[195,199],[19,22909,22911],{"className":22910},[204],[19,22912,22914,22934],{"className":22913},[208,209],[19,22915,22917,22931],{"className":22916},[213],[19,22918,22920],{"className":22919,"style":218},[217],[19,22921,22922,22925],{"style":221},[19,22923],{"className":22924,"style":226},[225],[19,22926,22928],{"className":22927},[230,231,232,233],[19,22929,52],{"className":22930},[195,199,233],[19,22932,241],{"className":22933},[240],[19,22935,22937],{"className":22936},[213],[19,22938,22940],{"className":22939,"style":248},[217],[19,22941],{},[19,22943],{"className":22944,"style":255},[254],[19,22946,56],{"className":22947},[259],[19,22949],{"className":22950,"style":255},[254],[19,22952,22954,22957,23006,23009,23012],{"className":22953},[186],[19,22955],{"className":22956,"style":1451},[190],[19,22958,22960,22963],{"className":22959},[195],[19,22961,49],{"className":22962,"style":200},[195,199],[19,22964,22966],{"className":22965},[204],[19,22967,22969,22998],{"className":22968},[208,209],[19,22970,22972,22995],{"className":22971},[213],[19,22973,22975],{"className":22974,"style":450},[217],[19,22976,22977,22980],{"style":221},[19,22978],{"className":22979,"style":226},[225],[19,22981,22983],{"className":22982},[230,231,232,233],[19,22984,22986,22989,22992],{"className":22985},[195,233],[19,22987,52],{"className":22988},[195,199,233],[19,22990,145],{"className":22991},[468,233],[19,22993,107],{"className":22994},[195,233],[19,22996,241],{"className":22997},[240],[19,22999,23001],{"className":23000},[213],[19,23002,23004],{"className":23003,"style":481},[217],[19,23005],{},[19,23007],{"className":23008,"style":487},[254],[19,23010,145],{"className":23011},[468],[19,23013],{"className":23014,"style":487},[254],[19,23016,23018,23021,23070,23073,23076],{"className":23017},[186],[19,23019],{"className":23020,"style":1451},[190],[19,23022,23024,23027],{"className":23023},[195],[19,23025,49],{"className":23026,"style":200},[195,199],[19,23028,23030],{"className":23029},[204],[19,23031,23033,23062],{"className":23032},[208,209],[19,23034,23036,23059],{"className":23035},[213],[19,23037,23039],{"className":23038,"style":450},[217],[19,23040,23041,23044],{"style":221},[19,23042],{"className":23043,"style":226},[225],[19,23045,23047],{"className":23046},[230,231,232,233],[19,23048,23050,23053,23056],{"className":23049},[195,233],[19,23051,52],{"className":23052},[195,199,233],[19,23054,145],{"className":23055},[468,233],[19,23057,107],{"className":23058},[195,233],[19,23060,241],{"className":23061},[240],[19,23063,23065],{"className":23064},[213],[19,23066,23068],{"className":23067,"style":481},[217],[19,23069],{},[19,23071],{"className":23072,"style":487},[254],[19,23074,140],{"className":23075},[468],[19,23077],{"className":23078,"style":487},[254],[19,23080,23082,23085,23088,23137,23140,23143],{"className":23081},[186],[19,23083],{"className":23084,"style":1451},[190],[19,23086,158],{"className":23087},[195],[19,23089,23091,23094],{"className":23090},[195],[19,23092,49],{"className":23093,"style":200},[195,199],[19,23095,23097],{"className":23096},[204],[19,23098,23100,23129],{"className":23099},[208,209],[19,23101,23103,23126],{"className":23102},[213],[19,23104,23106],{"className":23105,"style":450},[217],[19,23107,23108,23111],{"style":221},[19,23109],{"className":23110,"style":226},[225],[19,23112,23114],{"className":23113},[230,231,232,233],[19,23115,23117,23120,23123],{"className":23116},[195,233],[19,23118,52],{"className":23119},[195,199,233],[19,23121,145],{"className":23122},[468,233],[19,23124,107],{"className":23125},[195,233],[19,23127,241],{"className":23128},[240],[19,23130,23132],{"className":23131},[213],[19,23133,23135],{"className":23134,"style":481},[217],[19,23136],{},[19,23138],{"className":23139,"style":487},[254],[19,23141,145],{"className":23142},[468],[19,23144],{"className":23145,"style":487},[254],[19,23147,23149,23152,23192,23195,23198],{"className":23148},[186],[19,23150],{"className":23151,"style":191},[190],[19,23153,23155,23158],{"className":23154},[195],[19,23156,49],{"className":23157,"style":200},[195,199],[19,23159,23161],{"className":23160},[204],[19,23162,23164,23184],{"className":23163},[208,209],[19,23165,23167,23181],{"className":23166},[213],[19,23168,23170],{"className":23169,"style":218},[217],[19,23171,23172,23175],{"style":221},[19,23173],{"className":23174,"style":226},[225],[19,23176,23178],{"className":23177},[230,231,232,233],[19,23179,52],{"className":23180},[195,199,233],[19,23182,241],{"className":23183},[240],[19,23185,23187],{"className":23186},[213],[19,23188,23190],{"className":23189,"style":248},[217],[19,23191],{},[19,23193],{"className":23194,"style":255},[254],[19,23196,56],{"className":23197},[259],[19,23199],{"className":23200,"style":255},[254],[19,23202,23204,23207,23256,23259,23262],{"className":23203},[186],[19,23205],{"className":23206,"style":1451},[190],[19,23208,23210,23213],{"className":23209},[195],[19,23211,49],{"className":23212,"style":200},[195,199],[19,23214,23216],{"className":23215},[204],[19,23217,23219,23248],{"className":23218},[208,209],[19,23220,23222,23245],{"className":23221},[213],[19,23223,23225],{"className":23224,"style":450},[217],[19,23226,23227,23230],{"style":221},[19,23228],{"className":23229,"style":226},[225],[19,23231,23233],{"className":23232},[230,231,232,233],[19,23234,23236,23239,23242],{"className":23235},[195,233],[19,23237,52],{"className":23238},[195,199,233],[19,23240,145],{"className":23241},[468,233],[19,23243,107],{"className":23244},[195,233],[19,23246,241],{"className":23247},[240],[19,23249,23251],{"className":23250},[213],[19,23252,23254],{"className":23253,"style":481},[217],[19,23255],{},[19,23257],{"className":23258,"style":487},[254],[19,23260,140],{"className":23261},[468],[19,23263],{"className":23264,"style":487},[254],[19,23266,23268,23271,23311,23314,23317],{"className":23267},[186],[19,23269],{"className":23270,"style":191},[190],[19,23272,23274,23277],{"className":23273},[195],[19,23275,49],{"className":23276,"style":200},[195,199],[19,23278,23280],{"className":23279},[204],[19,23281,23283,23303],{"className":23282},[208,209],[19,23284,23286,23300],{"className":23285},[213],[19,23287,23289],{"className":23288,"style":218},[217],[19,23290,23291,23294],{"style":221},[19,23292],{"className":23293,"style":226},[225],[19,23295,23297],{"className":23296},[230,231,232,233],[19,23298,52],{"className":23299},[195,199,233],[19,23301,241],{"className":23302},[240],[19,23304,23306],{"className":23305},[213],[19,23307,23309],{"className":23308,"style":248},[217],[19,23310],{},[19,23312],{"className":23313,"style":255},[254],[19,23315,56],{"className":23316},[259],[19,23318],{"className":23319,"style":255},[254],[19,23321,23323,23326],{"className":23322},[186],[19,23324],{"className":23325,"style":191},[190],[19,23327,23329,23332],{"className":23328},[195],[19,23330,49],{"className":23331,"style":200},[195,199],[19,23333,23335],{"className":23334},[204],[19,23336,23338,23358],{"className":23337},[208,209],[19,23339,23341,23355],{"className":23340},[213],[19,23342,23344],{"className":23343,"style":218},[217],[19,23345,23346,23349],{"style":221},[19,23347],{"className":23348,"style":226},[225],[19,23350,23352],{"className":23351},[230,231,232,233],[19,23353,52],{"className":23354},[195,199,233],[19,23356,241],{"className":23357},[240],[19,23359,23361],{"className":23360},[213],[19,23362,23364],{"className":23363,"style":248},[217],[19,23365],{},[10,23367,23368,23369,14263],{},"Now, we can write the second-order linear homogeneous difference equation for ",[19,23370,23372,23390],{"className":23371,"translate":23},[27],[19,23373,23375],{"className":23374},[31],[33,23376,23377],{"xmlns":35},[38,23378,23379,23387],{},[41,23380,23381],{},[44,23382,23383,23385],{},[47,23384,49],{},[47,23386,52],{},[175,23388,23389],{"encoding":177},"F_n",[19,23391,23393],{"className":23392,"ariaHidden":61},[182],[19,23394,23396,23399],{"className":23395},[186],[19,23397],{"className":23398,"style":191},[190],[19,23400,23402,23405],{"className":23401},[195],[19,23403,49],{"className":23404,"style":200},[195,199],[19,23406,23408],{"className":23407},[204],[19,23409,23411,23431],{"className":23410},[208,209],[19,23412,23414,23428],{"className":23413},[213],[19,23415,23417],{"className":23416,"style":218},[217],[19,23418,23419,23422],{"style":221},[19,23420],{"className":23421,"style":226},[225],[19,23423,23425],{"className":23424},[230,231,232,233],[19,23426,52],{"className":23427},[195,199,233],[19,23429,241],{"className":23430},[240],[19,23432,23434],{"className":23433},[213],[19,23435,23437],{"className":23436,"style":248},[217],[19,23438],{},[19,23440,23442],{"className":23441,"translate":23},[22],[19,23443,23445,23481],{"className":23444,"translate":23},[27],[19,23446,23448],{"className":23447},[31],[33,23449,23450],{"xmlns":35,"display":36},[38,23451,23452,23478],{},[41,23453,23454,23460,23466,23468,23474,23476],{},[719,23455,23456,23458],{},[47,23457,20523],{"mathvariant":5770},[81,23459,158],{},[44,23461,23462,23464],{},[47,23463,49],{},[47,23465,52],{},[54,23467,140],{},[44,23469,23470,23472],{},[47,23471,49],{},[47,23473,52],{},[54,23475,56],{},[81,23477,78],{},[175,23479,23480],{"encoding":177},"\\Delta^2 F_n - F_n = 0",[19,23482,23484,23568,23623],{"className":23483,"ariaHidden":61},[182],[19,23485,23487,23490,23519,23559,23562,23565],{"className":23486},[186],[19,23488],{"className":23489,"style":20832},[190],[19,23491,23493,23496],{"className":23492},[195],[19,23494,20523],{"className":23495},[195],[19,23497,23499],{"className":23498},[204],[19,23500,23502],{"className":23501},[208],[19,23503,23505],{"className":23504},[213],[19,23506,23508],{"className":23507,"style":7748},[217],[19,23509,23510,23513],{"style":6430},[19,23511],{"className":23512,"style":226},[225],[19,23514,23516],{"className":23515},[230,231,232,233],[19,23517,158],{"className":23518},[195,233],[19,23520,23522,23525],{"className":23521},[195],[19,23523,49],{"className":23524,"style":200},[195,199],[19,23526,23528],{"className":23527},[204],[19,23529,23531,23551],{"className":23530},[208,209],[19,23532,23534,23548],{"className":23533},[213],[19,23535,23537],{"className":23536,"style":218},[217],[19,23538,23539,23542],{"style":221},[19,23540],{"className":23541,"style":226},[225],[19,23543,23545],{"className":23544},[230,231,232,233],[19,23546,52],{"className":23547},[195,199,233],[19,23549,241],{"className":23550},[240],[19,23552,23554],{"className":23553},[213],[19,23555,23557],{"className":23556,"style":248},[217],[19,23558],{},[19,23560],{"className":23561,"style":487},[254],[19,23563,140],{"className":23564},[468],[19,23566],{"className":23567,"style":487},[254],[19,23569,23571,23574,23614,23617,23620],{"className":23570},[186],[19,23572],{"className":23573,"style":191},[190],[19,23575,23577,23580],{"className":23576},[195],[19,23578,49],{"className":23579,"style":200},[195,199],[19,23581,23583],{"className":23582},[204],[19,23584,23586,23606],{"className":23585},[208,209],[19,23587,23589,23603],{"className":23588},[213],[19,23590,23592],{"className":23591,"style":218},[217],[19,23593,23594,23597],{"style":221},[19,23595],{"className":23596,"style":226},[225],[19,23598,23600],{"className":23599},[230,231,232,233],[19,23601,52],{"className":23602},[195,199,233],[19,23604,241],{"className":23605},[240],[19,23607,23609],{"className":23608},[213],[19,23610,23612],{"className":23611,"style":248},[217],[19,23613],{},[19,23615],{"className":23616,"style":255},[254],[19,23618,56],{"className":23619},[259],[19,23621],{"className":23622,"style":255},[254],[19,23624,23626,23629],{"className":23625},[186],[19,23627],{"className":23628,"style":1382},[190],[19,23630,78],{"className":23631},[195],[10,23633,23634,23635,14263],{},"This is a characteristic equation, and we can solve it by assuming ",[19,23636,23638,23665],{"className":23637,"translate":23},[27],[19,23639,23641],{"className":23640},[31],[33,23642,23643],{"xmlns":35},[38,23644,23645,23662],{},[41,23646,23647,23653,23655],{},[44,23648,23649,23651],{},[47,23650,49],{},[47,23652,52],{},[54,23654,56],{},[719,23656,23657,23660],{},[47,23658,23659],{},"r",[47,23661,52],{},[175,23663,23664],{"encoding":177},"F_n = r^n",[19,23666,23668,23723],{"className":23667,"ariaHidden":61},[182],[19,23669,23671,23674,23714,23717,23720],{"className":23670},[186],[19,23672],{"className":23673,"style":191},[190],[19,23675,23677,23680],{"className":23676},[195],[19,23678,49],{"className":23679,"style":200},[195,199],[19,23681,23683],{"className":23682},[204],[19,23684,23686,23706],{"className":23685},[208,209],[19,23687,23689,23703],{"className":23688},[213],[19,23690,23692],{"className":23691,"style":218},[217],[19,23693,23694,23697],{"style":221},[19,23695],{"className":23696,"style":226},[225],[19,23698,23700],{"className":23699},[230,231,232,233],[19,23701,52],{"className":23702},[195,199,233],[19,23704,241],{"className":23705},[240],[19,23707,23709],{"className":23708},[213],[19,23710,23712],{"className":23711,"style":248},[217],[19,23713],{},[19,23715],{"className":23716,"style":255},[254],[19,23718,56],{"className":23719},[259],[19,23721],{"className":23722,"style":255},[254],[19,23724,23726,23729],{"className":23725},[186],[19,23727],{"className":23728,"style":5443},[190],[19,23730,23732,23736],{"className":23731},[195],[19,23733,23659],{"className":23734,"style":23735},[195,199],"margin-right:0.02778em;",[19,23737,23739],{"className":23738},[204],[19,23740,23742],{"className":23741},[208],[19,23743,23745],{"className":23744},[213],[19,23746,23748],{"className":23747,"style":5443},[217],[19,23749,23750,23753],{"style":5446},[19,23751],{"className":23752,"style":226},[225],[19,23754,23756],{"className":23755},[230,231,232,233],[19,23757,52],{"className":23758},[195,199,233],[19,23760,23762],{"className":23761,"translate":23},[22],[19,23763,23765,23791],{"className":23764,"translate":23},[27],[19,23766,23768],{"className":23767},[31],[33,23769,23770],{"xmlns":35,"display":36},[38,23771,23772,23788],{},[41,23773,23774,23780,23782,23784,23786],{},[719,23775,23776,23778],{},[47,23777,23659],{},[81,23779,158],{},[54,23781,140],{},[81,23783,107],{},[54,23785,56],{},[81,23787,78],{},[175,23789,23790],{"encoding":177},"r^2 - 1 = 0",[19,23792,23794,23838,23856],{"className":23793,"ariaHidden":61},[182],[19,23795,23797,23800,23829,23832,23835],{"className":23796},[186],[19,23798],{"className":23799,"style":7729},[190],[19,23801,23803,23806],{"className":23802},[195],[19,23804,23659],{"className":23805,"style":23735},[195,199],[19,23807,23809],{"className":23808},[204],[19,23810,23812],{"className":23811},[208],[19,23813,23815],{"className":23814},[213],[19,23816,23818],{"className":23817,"style":7748},[217],[19,23819,23820,23823],{"style":6430},[19,23821],{"className":23822,"style":226},[225],[19,23824,23826],{"className":23825},[230,231,232,233],[19,23827,158],{"className":23828},[195,233],[19,23830],{"className":23831,"style":487},[254],[19,23833,140],{"className":23834},[468],[19,23836],{"className":23837,"style":487},[254],[19,23839,23841,23844,23847,23850,23853],{"className":23840},[186],[19,23842],{"className":23843,"style":1382},[190],[19,23845,107],{"className":23846},[195],[19,23848],{"className":23849,"style":255},[254],[19,23851,56],{"className":23852},[259],[19,23854],{"className":23855,"style":255},[254],[19,23857,23859,23862],{"className":23858},[186],[19,23860],{"className":23861,"style":1382},[190],[19,23863,78],{"className":23864},[195],[10,23866,23867,23868,5624,23919,23976],{},"Solving this equation gives us two solutions: ",[19,23869,23871,23889],{"className":23870,"translate":23},[27],[19,23872,23874],{"className":23873},[31],[33,23875,23876],{"xmlns":35},[38,23877,23878,23886],{},[41,23879,23880,23882,23884],{},[47,23881,23659],{},[54,23883,56],{},[81,23885,107],{},[175,23887,23888],{"encoding":177},"r = 1",[19,23890,23892,23910],{"className":23891,"ariaHidden":61},[182],[19,23893,23895,23898,23901,23904,23907],{"className":23894},[186],[19,23896],{"className":23897,"style":5104},[190],[19,23899,23659],{"className":23900,"style":23735},[195,199],[19,23902],{"className":23903,"style":255},[254],[19,23905,56],{"className":23906},[259],[19,23908],{"className":23909,"style":255},[254],[19,23911,23913,23916],{"className":23912},[186],[19,23914],{"className":23915,"style":1382},[190],[19,23917,107],{"className":23918},[195],[19,23920,23922,23942],{"className":23921,"translate":23},[27],[19,23923,23925],{"className":23924},[31],[33,23926,23927],{"xmlns":35},[38,23928,23929,23939],{},[41,23930,23931,23933,23935,23937],{},[47,23932,23659],{},[54,23934,56],{},[54,23936,140],{},[81,23938,107],{},[175,23940,23941],{"encoding":177},"r = -1",[19,23943,23945,23963],{"className":23944,"ariaHidden":61},[182],[19,23946,23948,23951,23954,23957,23960],{"className":23947},[186],[19,23949],{"className":23950,"style":5104},[190],[19,23952,23659],{"className":23953,"style":23735},[195,199],[19,23955],{"className":23956,"style":255},[254],[19,23958,56],{"className":23959},[259],[19,23961],{"className":23962,"style":255},[254],[19,23964,23966,23970,23973],{"className":23965},[186],[19,23967],{"className":23968,"style":23969},[190],"height:0.7278em;vertical-align:-0.0833em;",[19,23971,140],{"className":23972},[195],[19,23974,107],{"className":23975},[195],". Therefore, the general solution for the homogeneous difference equation is:",[19,23978,23980],{"className":23979,"translate":23},[22],[19,23981,23983,24040],{"className":23982,"translate":23},[27],[19,23984,23986],{"className":23985},[31],[33,23987,23988],{"xmlns":35,"display":36},[38,23989,23990,24037],{},[41,23991,23992,23998,24000,24007,24009,24015,24017,24023,24025,24027,24029,24031],{},[44,23993,23994,23996],{},[47,23995,49],{},[47,23997,52],{},[54,23999,56],{},[44,24001,24002,24005],{},[47,24003,24004],{},"c",[81,24006,107],{},[54,24008,1656],{},[719,24010,24011,24013],{},[81,24012,107],{},[47,24014,52],{},[54,24016,145],{},[44,24018,24019,24021],{},[47,24020,24004],{},[81,24022,158],{},[54,24024,1656],{},[54,24026,717],{"stretchy":79},[54,24028,140],{},[81,24030,107],{},[719,24032,24033,24035],{},[54,24034,741],{"stretchy":79},[47,24036,52],{},[175,24038,24039],{"encoding":177},"F_n = c_1 \\cdot 1^n + c_2 \\cdot (-1)^n",[19,24041,24043,24098,24154,24199,24254],{"className":24042,"ariaHidden":61},[182],[19,24044,24046,24049,24089,24092,24095],{"className":24045},[186],[19,24047],{"className":24048,"style":191},[190],[19,24050,24052,24055],{"className":24051},[195],[19,24053,49],{"className":24054,"style":200},[195,199],[19,24056,24058],{"className":24057},[204],[19,24059,24061,24081],{"className":24060},[208,209],[19,24062,24064,24078],{"className":24063},[213],[19,24065,24067],{"className":24066,"style":218},[217],[19,24068,24069,24072],{"style":221},[19,24070],{"className":24071,"style":226},[225],[19,24073,24075],{"className":24074},[230,231,232,233],[19,24076,52],{"className":24077},[195,199,233],[19,24079,241],{"className":24080},[240],[19,24082,24084],{"className":24083},[213],[19,24085,24087],{"className":24086,"style":248},[217],[19,24088],{},[19,24090],{"className":24091,"style":255},[254],[19,24093,56],{"className":24094},[259],[19,24096],{"className":24097,"style":255},[254],[19,24099,24101,24105,24145,24148,24151],{"className":24100},[186],[19,24102],{"className":24103,"style":24104},[190],"height:0.5945em;vertical-align:-0.15em;",[19,24106,24108,24111],{"className":24107},[195],[19,24109,24004],{"className":24110},[195,199],[19,24112,24114],{"className":24113},[204],[19,24115,24117,24137],{"className":24116},[208,209],[19,24118,24120,24134],{"className":24119},[213],[19,24121,24123],{"className":24122,"style":450},[217],[19,24124,24125,24128],{"style":5603},[19,24126],{"className":24127,"style":226},[225],[19,24129,24131],{"className":24130},[230,231,232,233],[19,24132,107],{"className":24133},[195,233],[19,24135,241],{"className":24136},[240],[19,24138,24140],{"className":24139},[213],[19,24141,24143],{"className":24142,"style":248},[217],[19,24144],{},[19,24146],{"className":24147,"style":487},[254],[19,24149,1656],{"className":24150},[468],[19,24152],{"className":24153,"style":487},[254],[19,24155,24157,24161,24190,24193,24196],{"className":24156},[186],[19,24158],{"className":24159,"style":24160},[190],"height:0.7977em;vertical-align:-0.0833em;",[19,24162,24164,24167],{"className":24163},[195],[19,24165,107],{"className":24166},[195],[19,24168,24170],{"className":24169},[204],[19,24171,24173],{"className":24172},[208],[19,24174,24176],{"className":24175},[213],[19,24177,24179],{"className":24178,"style":6427},[217],[19,24180,24181,24184],{"style":6430},[19,24182],{"className":24183,"style":226},[225],[19,24185,24187],{"className":24186},[230,231,232,233],[19,24188,52],{"className":24189},[195,199,233],[19,24191],{"className":24192,"style":487},[254],[19,24194,145],{"className":24195},[468],[19,24197],{"className":24198,"style":487},[254],[19,24200,24202,24205,24245,24248,24251],{"className":24201},[186],[19,24203],{"className":24204,"style":24104},[190],[19,24206,24208,24211],{"className":24207},[195],[19,24209,24004],{"className":24210},[195,199],[19,24212,24214],{"className":24213},[204],[19,24215,24217,24237],{"className":24216},[208,209],[19,24218,24220,24234],{"className":24219},[213],[19,24221,24223],{"className":24222,"style":450},[217],[19,24224,24225,24228],{"style":5603},[19,24226],{"className":24227,"style":226},[225],[19,24229,24231],{"className":24230},[230,231,232,233],[19,24232,158],{"className":24233},[195,233],[19,24235,241],{"className":24236},[240],[19,24238,24240],{"className":24239},[213],[19,24241,24243],{"className":24242,"style":248},[217],[19,24244],{},[19,24246],{"className":24247,"style":487},[254],[19,24249,1656],{"className":24250},[468],[19,24252],{"className":24253,"style":487},[254],[19,24255,24257,24260,24263,24266,24269],{"className":24256},[186],[19,24258],{"className":24259,"style":5312},[190],[19,24261,717],{"className":24262},[277],[19,24264,140],{"className":24265},[195],[19,24267,107],{"className":24268},[195],[19,24270,24272,24275],{"className":24271},[674],[19,24273,741],{"className":24274},[674],[19,24276,24278],{"className":24277},[204],[19,24279,24281],{"className":24280},[208],[19,24282,24284],{"className":24283},[213],[19,24285,24287],{"className":24286,"style":6427},[217],[19,24288,24289,24292],{"style":6430},[19,24290],{"className":24291,"style":226},[225],[19,24293,24295],{"className":24294},[230,231,232,233],[19,24296,52],{"className":24297},[195,199,233],[10,24299,24300,24301,5624,24393,24485],{},"Now, we need initial conditions to find the particular solution. We know that ",[19,24302,24304,24326],{"className":24303,"translate":23},[27],[19,24305,24307],{"className":24306},[31],[33,24308,24309],{"xmlns":35},[38,24310,24311,24323],{},[41,24312,24313,24319,24321],{},[44,24314,24315,24317],{},[47,24316,49],{},[81,24318,78],{},[54,24320,56],{},[81,24322,78],{},[175,24324,24325],{"encoding":177},"F_0 = 0",[19,24327,24329,24384],{"className":24328,"ariaHidden":61},[182],[19,24330,24332,24335,24375,24378,24381],{"className":24331},[186],[19,24333],{"className":24334,"style":191},[190],[19,24336,24338,24341],{"className":24337},[195],[19,24339,49],{"className":24340,"style":200},[195,199],[19,24342,24344],{"className":24343},[204],[19,24345,24347,24367],{"className":24346},[208,209],[19,24348,24350,24364],{"className":24349},[213],[19,24351,24353],{"className":24352,"style":450},[217],[19,24354,24355,24358],{"style":221},[19,24356],{"className":24357,"style":226},[225],[19,24359,24361],{"className":24360},[230,231,232,233],[19,24362,78],{"className":24363},[195,233],[19,24365,241],{"className":24366},[240],[19,24368,24370],{"className":24369},[213],[19,24371,24373],{"className":24372,"style":248},[217],[19,24374],{},[19,24376],{"className":24377,"style":255},[254],[19,24379,56],{"className":24380},[259],[19,24382],{"className":24383,"style":255},[254],[19,24385,24387,24390],{"className":24386},[186],[19,24388],{"className":24389,"style":1382},[190],[19,24391,78],{"className":24392},[195],[19,24394,24396,24418],{"className":24395,"translate":23},[27],[19,24397,24399],{"className":24398},[31],[33,24400,24401],{"xmlns":35},[38,24402,24403,24415],{},[41,24404,24405,24411,24413],{},[44,24406,24407,24409],{},[47,24408,49],{},[81,24410,107],{},[54,24412,56],{},[81,24414,107],{},[175,24416,24417],{"encoding":177},"F_1 = 1",[19,24419,24421,24476],{"className":24420,"ariaHidden":61},[182],[19,24422,24424,24427,24467,24470,24473],{"className":24423},[186],[19,24425],{"className":24426,"style":191},[190],[19,24428,24430,24433],{"className":24429},[195],[19,24431,49],{"className":24432,"style":200},[195,199],[19,24434,24436],{"className":24435},[204],[19,24437,24439,24459],{"className":24438},[208,209],[19,24440,24442,24456],{"className":24441},[213],[19,24443,24445],{"className":24444,"style":450},[217],[19,24446,24447,24450],{"style":221},[19,24448],{"className":24449,"style":226},[225],[19,24451,24453],{"className":24452},[230,231,232,233],[19,24454,107],{"className":24455},[195,233],[19,24457,241],{"className":24458},[240],[19,24460,24462],{"className":24461},[213],[19,24463,24465],{"className":24464,"style":248},[217],[19,24466],{},[19,24468],{"className":24469,"style":255},[254],[19,24471,56],{"className":24472},[259],[19,24474],{"className":24475,"style":255},[254],[19,24477,24479,24482],{"className":24478},[186],[19,24480],{"className":24481,"style":1382},[190],[19,24483,107],{"className":24484},[195],". Substituting these into the general solution:",[19,24487,24489],{"className":24488,"translate":23},[22],[19,24490,24492,24660],{"className":24491,"translate":23},[27],[19,24493,24495],{"className":24494},[31],[33,24496,24497],{"xmlns":35,"display":36},[38,24498,24499,24657],{},[64,24500,24501,24579],{"rowspacing":3039,"columnalign":3040,"columnspacing":3041},[70,24502,24503,24513],{},[73,24504,24505],{},[76,24506,24507],{"scriptlevel":78,"displaystyle":61},[44,24508,24509,24511],{},[47,24510,49],{},[81,24512,78],{},[73,24514,24515],{},[76,24516,24517],{"scriptlevel":78,"displaystyle":61},[41,24518,24519,24521,24523,24529,24531,24537,24539,24545,24547,24549,24551,24553,24559,24561,24567,24569,24575,24577],{},[41,24520],{},[54,24522,56],{},[44,24524,24525,24527],{},[47,24526,24004],{},[81,24528,107],{},[54,24530,1656],{},[719,24532,24533,24535],{},[81,24534,107],{},[81,24536,78],{},[54,24538,145],{},[44,24540,24541,24543],{},[47,24542,24004],{},[81,24544,158],{},[54,24546,1656],{},[54,24548,717],{"stretchy":79},[54,24550,140],{},[81,24552,107],{},[719,24554,24555,24557],{},[54,24556,741],{"stretchy":79},[81,24558,78],{},[54,24560,56],{},[44,24562,24563,24565],{},[47,24564,24004],{},[81,24566,107],{},[54,24568,145],{},[44,24570,24571,24573],{},[47,24572,24004],{},[81,24574,158],{},[54,24576,56],{},[81,24578,78],{},[70,24580,24581,24591],{},[73,24582,24583],{},[76,24584,24585],{"scriptlevel":78,"displaystyle":61},[44,24586,24587,24589],{},[47,24588,49],{},[81,24590,107],{},[73,24592,24593],{},[76,24594,24595],{"scriptlevel":78,"displaystyle":61},[41,24596,24597,24599,24601,24607,24609,24615,24617,24623,24625,24627,24629,24631,24637,24639,24645,24647,24653,24655],{},[41,24598],{},[54,24600,56],{},[44,24602,24603,24605],{},[47,24604,24004],{},[81,24606,107],{},[54,24608,1656],{},[719,24610,24611,24613],{},[81,24612,107],{},[81,24614,107],{},[54,24616,145],{},[44,24618,24619,24621],{},[47,24620,24004],{},[81,24622,158],{},[54,24624,1656],{},[54,24626,717],{"stretchy":79},[54,24628,140],{},[81,24630,107],{},[719,24632,24633,24635],{},[54,24634,741],{"stretchy":79},[81,24636,107],{},[54,24638,56],{},[44,24640,24641,24643],{},[47,24642,24004],{},[81,24644,107],{},[54,24646,140],{},[44,24648,24649,24651],{},[47,24650,24004],{},[81,24652,158],{},[54,24654,56],{},[81,24656,107],{},[175,24658,24659],{"encoding":177},"\\begin{align*}\nF_0 &= c_1 \\cdot 1^0 + c_2 \\cdot (-1)^0 = c_1 + c_2 = 0 \\\\\nF_1 &= c_1 \\cdot 1^1 + c_2 \\cdot (-1)^1 = c_1 - c_2 = 1\n\\end{align*}",[19,24661,24663],{"className":24662,"ariaHidden":61},[182],[19,24664,24666,24670],{"className":24665},[186],[19,24667],{"className":24668,"style":24669},[190],"height:3.0482em;vertical-align:-1.2741em;",[19,24671,24673],{"className":24672},[195],[19,24674,24676,24799],{"className":24675},[64],[19,24677,24679],{"className":24678},[3453],[19,24680,24682,24790],{"className":24681},[208,209],[19,24683,24685,24787],{"className":24684},[213],[19,24686,24689,24738],{"className":24687,"style":24688},[217],"height:1.7741em;",[19,24690,24692,24695],{"style":24691},"top:-3.91em;",[19,24693],{"className":24694,"style":865},[225],[19,24696,24698],{"className":24697},[195],[19,24699,24701,24704],{"className":24700},[195],[19,24702,49],{"className":24703,"style":200},[195,199],[19,24705,24707],{"className":24706},[204],[19,24708,24710,24730],{"className":24709},[208,209],[19,24711,24713,24727],{"className":24712},[213],[19,24714,24716],{"className":24715,"style":450},[217],[19,24717,24718,24721],{"style":221},[19,24719],{"className":24720,"style":226},[225],[19,24722,24724],{"className":24723},[230,231,232,233],[19,24725,78],{"className":24726},[195,233],[19,24728,241],{"className":24729},[240],[19,24731,24733],{"className":24732},[213],[19,24734,24736],{"className":24735,"style":248},[217],[19,24737],{},[19,24739,24741,24744],{"style":24740},"top:-2.3859em;",[19,24742],{"className":24743,"style":865},[225],[19,24745,24747],{"className":24746},[195],[19,24748,24750,24753],{"className":24749},[195],[19,24751,49],{"className":24752,"style":200},[195,199],[19,24754,24756],{"className":24755},[204],[19,24757,24759,24779],{"className":24758},[208,209],[19,24760,24762,24776],{"className":24761},[213],[19,24763,24765],{"className":24764,"style":450},[217],[19,24766,24767,24770],{"style":221},[19,24768],{"className":24769,"style":226},[225],[19,24771,24773],{"className":24772},[230,231,232,233],[19,24774,107],{"className":24775},[195,233],[19,24777,241],{"className":24778},[240],[19,24780,24782],{"className":24781},[213],[19,24783,24785],{"className":24784,"style":248},[217],[19,24786],{},[19,24788,241],{"className":24789},[240],[19,24791,24793],{"className":24792},[213],[19,24794,24797],{"className":24795,"style":24796},[217],"height:1.2741em;",[19,24798],{},[19,24800,24802],{"className":24801},[387],[19,24803,24805,25422],{"className":24804},[208,209],[19,24806,24808,25419],{"className":24807},[213],[19,24809,24811,25115],{"className":24810,"style":24688},[217],[19,24812,24813,24816],{"style":24691},[19,24814],{"className":24815,"style":865},[225],[19,24817,24819,24822,24825,24828,24831,24871,24874,24877,24880,24909,24912,24915,24918,24958,24961,24964,24967,24970,24973,24976,25005,25008,25011,25014,25054,25057,25060,25063,25103,25106,25109,25112],{"className":24818},[195],[19,24820],{"className":24821},[195],[19,24823],{"className":24824,"style":255},[254],[19,24826,56],{"className":24827},[259],[19,24829],{"className":24830,"style":255},[254],[19,24832,24834,24837],{"className":24833},[195],[19,24835,24004],{"className":24836},[195,199],[19,24838,24840],{"className":24839},[204],[19,24841,24843,24863],{"className":24842},[208,209],[19,24844,24846,24860],{"className":24845},[213],[19,24847,24849],{"className":24848,"style":450},[217],[19,24850,24851,24854],{"style":5603},[19,24852],{"className":24853,"style":226},[225],[19,24855,24857],{"className":24856},[230,231,232,233],[19,24858,107],{"className":24859},[195,233],[19,24861,241],{"className":24862},[240],[19,24864,24866],{"className":24865},[213],[19,24867,24869],{"className":24868,"style":248},[217],[19,24870],{},[19,24872],{"className":24873,"style":487},[254],[19,24875,1656],{"className":24876},[468],[19,24878],{"className":24879,"style":487},[254],[19,24881,24883,24886],{"className":24882},[195],[19,24884,107],{"className":24885},[195],[19,24887,24889],{"className":24888},[204],[19,24890,24892],{"className":24891},[208],[19,24893,24895],{"className":24894},[213],[19,24896,24898],{"className":24897,"style":7748},[217],[19,24899,24900,24903],{"style":6430},[19,24901],{"className":24902,"style":226},[225],[19,24904,24906],{"className":24905},[230,231,232,233],[19,24907,78],{"className":24908},[195,233],[19,24910],{"className":24911,"style":487},[254],[19,24913,145],{"className":24914},[468],[19,24916],{"className":24917,"style":487},[254],[19,24919,24921,24924],{"className":24920},[195],[19,24922,24004],{"className":24923},[195,199],[19,24925,24927],{"className":24926},[204],[19,24928,24930,24950],{"className":24929},[208,209],[19,24931,24933,24947],{"className":24932},[213],[19,24934,24936],{"className":24935,"style":450},[217],[19,24937,24938,24941],{"style":5603},[19,24939],{"className":24940,"style":226},[225],[19,24942,24944],{"className":24943},[230,231,232,233],[19,24945,158],{"className":24946},[195,233],[19,24948,241],{"className":24949},[240],[19,24951,24953],{"className":24952},[213],[19,24954,24956],{"className":24955,"style":248},[217],[19,24957],{},[19,24959],{"className":24960,"style":487},[254],[19,24962,1656],{"className":24963},[468],[19,24965],{"className":24966,"style":487},[254],[19,24968,717],{"className":24969},[277],[19,24971,140],{"className":24972},[195],[19,24974,107],{"className":24975},[195],[19,24977,24979,24982],{"className":24978},[674],[19,24980,741],{"className":24981},[674],[19,24983,24985],{"className":24984},[204],[19,24986,24988],{"className":24987},[208],[19,24989,24991],{"className":24990},[213],[19,24992,24994],{"className":24993,"style":7748},[217],[19,24995,24996,24999],{"style":6430},[19,24997],{"className":24998,"style":226},[225],[19,25000,25002],{"className":25001},[230,231,232,233],[19,25003,78],{"className":25004},[195,233],[19,25006],{"className":25007,"style":255},[254],[19,25009,56],{"className":25010},[259],[19,25012],{"className":25013,"style":255},[254],[19,25015,25017,25020],{"className":25016},[195],[19,25018,24004],{"className":25019},[195,199],[19,25021,25023],{"className":25022},[204],[19,25024,25026,25046],{"className":25025},[208,209],[19,25027,25029,25043],{"className":25028},[213],[19,25030,25032],{"className":25031,"style":450},[217],[19,25033,25034,25037],{"style":5603},[19,25035],{"className":25036,"style":226},[225],[19,25038,25040],{"className":25039},[230,231,232,233],[19,25041,107],{"className":25042},[195,233],[19,25044,241],{"className":25045},[240],[19,25047,25049],{"className":25048},[213],[19,25050,25052],{"className":25051,"style":248},[217],[19,25053],{},[19,25055],{"className":25056,"style":487},[254],[19,25058,145],{"className":25059},[468],[19,25061],{"className":25062,"style":487},[254],[19,25064,25066,25069],{"className":25065},[195],[19,25067,24004],{"className":25068},[195,199],[19,25070,25072],{"className":25071},[204],[19,25073,25075,25095],{"className":25074},[208,209],[19,25076,25078,25092],{"className":25077},[213],[19,25079,25081],{"className":25080,"style":450},[217],[19,25082,25083,25086],{"style":5603},[19,25084],{"className":25085,"style":226},[225],[19,25087,25089],{"className":25088},[230,231,232,233],[19,25090,158],{"className":25091},[195,233],[19,25093,241],{"className":25094},[240],[19,25096,25098],{"className":25097},[213],[19,25099,25101],{"className":25100,"style":248},[217],[19,25102],{},[19,25104],{"className":25105,"style":255},[254],[19,25107,56],{"className":25108},[259],[19,25110],{"className":25111,"style":255},[254],[19,25113,78],{"className":25114},[195],[19,25116,25117,25120],{"style":24740},[19,25118],{"className":25119,"style":865},[225],[19,25121,25123,25126,25129,25132,25135,25175,25178,25181,25184,25213,25216,25219,25222,25262,25265,25268,25271,25274,25277,25280,25309,25312,25315,25318,25358,25361,25364,25367,25407,25410,25413,25416],{"className":25122},[195],[19,25124],{"className":25125},[195],[19,25127],{"className":25128,"style":255},[254],[19,25130,56],{"className":25131},[259],[19,25133],{"className":25134,"style":255},[254],[19,25136,25138,25141],{"className":25137},[195],[19,25139,24004],{"className":25140},[195,199],[19,25142,25144],{"className":25143},[204],[19,25145,25147,25167],{"className":25146},[208,209],[19,25148,25150,25164],{"className":25149},[213],[19,25151,25153],{"className":25152,"style":450},[217],[19,25154,25155,25158],{"style":5603},[19,25156],{"className":25157,"style":226},[225],[19,25159,25161],{"className":25160},[230,231,232,233],[19,25162,107],{"className":25163},[195,233],[19,25165,241],{"className":25166},[240],[19,25168,25170],{"className":25169},[213],[19,25171,25173],{"className":25172,"style":248},[217],[19,25174],{},[19,25176],{"className":25177,"style":487},[254],[19,25179,1656],{"className":25180},[468],[19,25182],{"className":25183,"style":487},[254],[19,25185,25187,25190],{"className":25186},[195],[19,25188,107],{"className":25189},[195],[19,25191,25193],{"className":25192},[204],[19,25194,25196],{"className":25195},[208],[19,25197,25199],{"className":25198},[213],[19,25200,25202],{"className":25201,"style":7748},[217],[19,25203,25204,25207],{"style":6430},[19,25205],{"className":25206,"style":226},[225],[19,25208,25210],{"className":25209},[230,231,232,233],[19,25211,107],{"className":25212},[195,233],[19,25214],{"className":25215,"style":487},[254],[19,25217,145],{"className":25218},[468],[19,25220],{"className":25221,"style":487},[254],[19,25223,25225,25228],{"className":25224},[195],[19,25226,24004],{"className":25227},[195,199],[19,25229,25231],{"className":25230},[204],[19,25232,25234,25254],{"className":25233},[208,209],[19,25235,25237,25251],{"className":25236},[213],[19,25238,25240],{"className":25239,"style":450},[217],[19,25241,25242,25245],{"style":5603},[19,25243],{"className":25244,"style":226},[225],[19,25246,25248],{"className":25247},[230,231,232,233],[19,25249,158],{"className":25250},[195,233],[19,25252,241],{"className":25253},[240],[19,25255,25257],{"className":25256},[213],[19,25258,25260],{"className":25259,"style":248},[217],[19,25261],{},[19,25263],{"className":25264,"style":487},[254],[19,25266,1656],{"className":25267},[468],[19,25269],{"className":25270,"style":487},[254],[19,25272,717],{"className":25273},[277],[19,25275,140],{"className":25276},[195],[19,25278,107],{"className":25279},[195],[19,25281,25283,25286],{"className":25282},[674],[19,25284,741],{"className":25285},[674],[19,25287,25289],{"className":25288},[204],[19,25290,25292],{"className":25291},[208],[19,25293,25295],{"className":25294},[213],[19,25296,25298],{"className":25297,"style":7748},[217],[19,25299,25300,25303],{"style":6430},[19,25301],{"className":25302,"style":226},[225],[19,25304,25306],{"className":25305},[230,231,232,233],[19,25307,107],{"className":25308},[195,233],[19,25310],{"className":25311,"style":255},[254],[19,25313,56],{"className":25314},[259],[19,25316],{"className":25317,"style":255},[254],[19,25319,25321,25324],{"className":25320},[195],[19,25322,24004],{"className":25323},[195,199],[19,25325,25327],{"className":25326},[204],[19,25328,25330,25350],{"className":25329},[208,209],[19,25331,25333,25347],{"className":25332},[213],[19,25334,25336],{"className":25335,"style":450},[217],[19,25337,25338,25341],{"style":5603},[19,25339],{"className":25340,"style":226},[225],[19,25342,25344],{"className":25343},[230,231,232,233],[19,25345,107],{"className":25346},[195,233],[19,25348,241],{"className":25349},[240],[19,25351,25353],{"className":25352},[213],[19,25354,25356],{"className":25355,"style":248},[217],[19,25357],{},[19,25359],{"className":25360,"style":487},[254],[19,25362,140],{"className":25363},[468],[19,25365],{"className":25366,"style":487},[254],[19,25368,25370,25373],{"className":25369},[195],[19,25371,24004],{"className":25372},[195,199],[19,25374,25376],{"className":25375},[204],[19,25377,25379,25399],{"className":25378},[208,209],[19,25380,25382,25396],{"className":25381},[213],[19,25383,25385],{"className":25384,"style":450},[217],[19,25386,25387,25390],{"style":5603},[19,25388],{"className":25389,"style":226},[225],[19,25391,25393],{"className":25392},[230,231,232,233],[19,25394,158],{"className":25395},[195,233],[19,25397,241],{"className":25398},[240],[19,25400,25402],{"className":25401},[213],[19,25403,25405],{"className":25404,"style":248},[217],[19,25406],{},[19,25408],{"className":25409,"style":255},[254],[19,25411,56],{"className":25412},[259],[19,25414],{"className":25415,"style":255},[254],[19,25417,107],{"className":25418},[195],[19,25420,241],{"className":25421},[240],[19,25423,25425],{"className":25424},[213],[19,25426,25428],{"className":25427,"style":24796},[217],[19,25429],{},[10,25431,25432,25433,5624,25597,25763],{},"Solving this system of equations, we find ",[19,25434,25436,25462],{"className":25435,"translate":23},[27],[19,25437,25439],{"className":25438},[31],[33,25440,25441],{"xmlns":35},[38,25442,25443,25459],{},[41,25444,25445,25451,25453],{},[44,25446,25447,25449],{},[47,25448,24004],{},[81,25450,107],{},[54,25452,56],{},[703,25454,25455,25457],{},[81,25456,107],{},[81,25458,158],{},[175,25460,25461],{"encoding":177},"c_1 = \\frac{1}{2}",[19,25463,25465,25520],{"className":25464,"ariaHidden":61},[182],[19,25466,25468,25471,25511,25514,25517],{"className":25467},[186],[19,25469],{"className":25470,"style":20687},[190],[19,25472,25474,25477],{"className":25473},[195],[19,25475,24004],{"className":25476},[195,199],[19,25478,25480],{"className":25479},[204],[19,25481,25483,25503],{"className":25482},[208,209],[19,25484,25486,25500],{"className":25485},[213],[19,25487,25489],{"className":25488,"style":450},[217],[19,25490,25491,25494],{"style":5603},[19,25492],{"className":25493,"style":226},[225],[19,25495,25497],{"className":25496},[230,231,232,233],[19,25498,107],{"className":25499},[195,233],[19,25501,241],{"className":25502},[240],[19,25504,25506],{"className":25505},[213],[19,25507,25509],{"className":25508,"style":248},[217],[19,25510],{},[19,25512],{"className":25513,"style":255},[254],[19,25515,56],{"className":25516},[259],[19,25518],{"className":25519,"style":255},[254],[19,25521,25523,25527],{"className":25522},[186],[19,25524],{"className":25525,"style":25526},[190],"height:1.1901em;vertical-align:-0.345em;",[19,25528,25530,25533,25594],{"className":25529},[195],[19,25531],{"className":25532},[277,675],[19,25534,25536],{"className":25535},[703],[19,25537,25539,25586],{"className":25538},[208,209],[19,25540,25542,25583],{"className":25541},[213],[19,25543,25546,25560,25568],{"className":25544,"style":25545},[217],"height:0.8451em;",[19,25547,25548,25551],{"style":7941},[19,25549],{"className":25550,"style":865},[225],[19,25552,25554],{"className":25553},[230,231,232,233],[19,25555,25557],{"className":25556},[195,233],[19,25558,158],{"className":25559},[195,233],[19,25561,25562,25565],{"style":932},[19,25563],{"className":25564,"style":865},[225],[19,25566],{"className":25567,"style":940},[939],[19,25569,25571,25574],{"style":25570},"top:-3.394em;",[19,25572],{"className":25573,"style":865},[225],[19,25575,25577],{"className":25576},[230,231,232,233],[19,25578,25580],{"className":25579},[195,233],[19,25581,107],{"className":25582},[195,233],[19,25584,241],{"className":25585},[240],[19,25587,25589],{"className":25588},[213],[19,25590,25592],{"className":25591,"style":8039},[217],[19,25593],{},[19,25595],{"className":25596},[674,675],[19,25598,25600,25628],{"className":25599,"translate":23},[27],[19,25601,25603],{"className":25602},[31],[33,25604,25605],{"xmlns":35},[38,25606,25607,25625],{},[41,25608,25609,25615,25617,25619],{},[44,25610,25611,25613],{},[47,25612,24004],{},[81,25614,158],{},[54,25616,56],{},[54,25618,140],{},[703,25620,25621,25623],{},[81,25622,107],{},[81,25624,158],{},[175,25626,25627],{"encoding":177},"c_2 = -\\frac{1}{2}",[19,25629,25631,25686],{"className":25630,"ariaHidden":61},[182],[19,25632,25634,25637,25677,25680,25683],{"className":25633},[186],[19,25635],{"className":25636,"style":20687},[190],[19,25638,25640,25643],{"className":25639},[195],[19,25641,24004],{"className":25642},[195,199],[19,25644,25646],{"className":25645},[204],[19,25647,25649,25669],{"className":25648},[208,209],[19,25650,25652,25666],{"className":25651},[213],[19,25653,25655],{"className":25654,"style":450},[217],[19,25656,25657,25660],{"style":5603},[19,25658],{"className":25659,"style":226},[225],[19,25661,25663],{"className":25662},[230,231,232,233],[19,25664,158],{"className":25665},[195,233],[19,25667,241],{"className":25668},[240],[19,25670,25672],{"className":25671},[213],[19,25673,25675],{"className":25674,"style":248},[217],[19,25676],{},[19,25678],{"className":25679,"style":255},[254],[19,25681,56],{"className":25682},[259],[19,25684],{"className":25685,"style":255},[254],[19,25687,25689,25692,25695],{"className":25688},[186],[19,25690],{"className":25691,"style":25526},[190],[19,25693,140],{"className":25694},[195],[19,25696,25698,25701,25760],{"className":25697},[195],[19,25699],{"className":25700},[277,675],[19,25702,25704],{"className":25703},[703],[19,25705,25707,25752],{"className":25706},[208,209],[19,25708,25710,25749],{"className":25709},[213],[19,25711,25713,25727,25735],{"className":25712,"style":25545},[217],[19,25714,25715,25718],{"style":7941},[19,25716],{"className":25717,"style":865},[225],[19,25719,25721],{"className":25720},[230,231,232,233],[19,25722,25724],{"className":25723},[195,233],[19,25725,158],{"className":25726},[195,233],[19,25728,25729,25732],{"style":932},[19,25730],{"className":25731,"style":865},[225],[19,25733],{"className":25734,"style":940},[939],[19,25736,25737,25740],{"style":25570},[19,25738],{"className":25739,"style":865},[225],[19,25741,25743],{"className":25742},[230,231,232,233],[19,25744,25746],{"className":25745},[195,233],[19,25747,107],{"className":25748},[195,233],[19,25750,241],{"className":25751},[240],[19,25753,25755],{"className":25754},[213],[19,25756,25758],{"className":25757,"style":8039},[217],[19,25759],{},[19,25761],{"className":25762},[674,675],". Therefore, the particular solution for the Fibonacci sequence is:",[19,25765,25767],{"className":25766,"translate":23},[22],[19,25768,25770,25826],{"className":25769,"translate":23},[27],[19,25771,25773],{"className":25772},[31],[33,25774,25775],{"xmlns":35,"display":36},[38,25776,25777,25823],{},[41,25778,25779,25785,25787,25793,25795,25801,25803,25809,25811,25813,25815,25817],{},[44,25780,25781,25783],{},[47,25782,49],{},[47,25784,52],{},[54,25786,56],{},[703,25788,25789,25791],{},[81,25790,107],{},[81,25792,158],{},[54,25794,1656],{},[719,25796,25797,25799],{},[81,25798,107],{},[47,25800,52],{},[54,25802,140],{},[703,25804,25805,25807],{},[81,25806,107],{},[81,25808,158],{},[54,25810,1656],{},[54,25812,717],{"stretchy":79},[54,25814,140],{},[81,25816,107],{},[719,25818,25819,25821],{},[54,25820,741],{"stretchy":79},[47,25822,52],{},[175,25824,25825],{"encoding":177},"F_n = \\frac{1}{2} \\cdot 1^n - \\frac{1}{2} \\cdot (-1)^n",[19,25827,25829,25884,25962,26006,26083],{"className":25828,"ariaHidden":61},[182],[19,25830,25832,25835,25875,25878,25881],{"className":25831},[186],[19,25833],{"className":25834,"style":191},[190],[19,25836,25838,25841],{"className":25837},[195],[19,25839,49],{"className":25840,"style":200},[195,199],[19,25842,25844],{"className":25843},[204],[19,25845,25847,25867],{"className":25846},[208,209],[19,25848,25850,25864],{"className":25849},[213],[19,25851,25853],{"className":25852,"style":218},[217],[19,25854,25855,25858],{"style":221},[19,25856],{"className":25857,"style":226},[225],[19,25859,25861],{"className":25860},[230,231,232,233],[19,25862,52],{"className":25863},[195,199,233],[19,25865,241],{"className":25866},[240],[19,25868,25870],{"className":25869},[213],[19,25871,25873],{"className":25872,"style":248},[217],[19,25874],{},[19,25876],{"className":25877,"style":255},[254],[19,25879,56],{"className":25880},[259],[19,25882],{"className":25883,"style":255},[254],[19,25885,25887,25891,25953,25956,25959],{"className":25886},[186],[19,25888],{"className":25889,"style":25890},[190],"height:2.0074em;vertical-align:-0.686em;",[19,25892,25894,25897,25950],{"className":25893},[195],[19,25895],{"className":25896},[277,675],[19,25898,25900],{"className":25899},[703],[19,25901,25903,25942],{"className":25902},[208,209],[19,25904,25906,25939],{"className":25905},[213],[19,25907,25909,25920,25928],{"className":25908,"style":858},[217],[19,25910,25911,25914],{"style":1017},[19,25912],{"className":25913,"style":865},[225],[19,25915,25917],{"className":25916},[195],[19,25918,158],{"className":25919},[195],[19,25921,25922,25925],{"style":932},[19,25923],{"className":25924,"style":865},[225],[19,25926],{"className":25927,"style":940},[939],[19,25929,25930,25933],{"style":943},[19,25931],{"className":25932,"style":865},[225],[19,25934,25936],{"className":25935},[195],[19,25937,107],{"className":25938},[195],[19,25940,241],{"className":25941},[240],[19,25943,25945],{"className":25944},[213],[19,25946,25948],{"className":25947,"style":1111},[217],[19,25949],{},[19,25951],{"className":25952},[674,675],[19,25954],{"className":25955,"style":487},[254],[19,25957,1656],{"className":25958},[468],[19,25960],{"className":25961,"style":487},[254],[19,25963,25965,25968,25997,26000,26003],{"className":25964},[186],[19,25966],{"className":25967,"style":24160},[190],[19,25969,25971,25974],{"className":25970},[195],[19,25972,107],{"className":25973},[195],[19,25975,25977],{"className":25976},[204],[19,25978,25980],{"className":25979},[208],[19,25981,25983],{"className":25982},[213],[19,25984,25986],{"className":25985,"style":6427},[217],[19,25987,25988,25991],{"style":6430},[19,25989],{"className":25990,"style":226},[225],[19,25992,25994],{"className":25993},[230,231,232,233],[19,25995,52],{"className":25996},[195,199,233],[19,25998],{"className":25999,"style":487},[254],[19,26001,140],{"className":26002},[468],[19,26004],{"className":26005,"style":487},[254],[19,26007,26009,26012,26074,26077,26080],{"className":26008},[186],[19,26010],{"className":26011,"style":25890},[190],[19,26013,26015,26018,26071],{"className":26014},[195],[19,26016],{"className":26017},[277,675],[19,26019,26021],{"className":26020},[703],[19,26022,26024,26063],{"className":26023},[208,209],[19,26025,26027,26060],{"className":26026},[213],[19,26028,26030,26041,26049],{"className":26029,"style":858},[217],[19,26031,26032,26035],{"style":1017},[19,26033],{"className":26034,"style":865},[225],[19,26036,26038],{"className":26037},[195],[19,26039,158],{"className":26040},[195],[19,26042,26043,26046],{"style":932},[19,26044],{"className":26045,"style":865},[225],[19,26047],{"className":26048,"style":940},[939],[19,26050,26051,26054],{"style":943},[19,26052],{"className":26053,"style":865},[225],[19,26055,26057],{"className":26056},[195],[19,26058,107],{"className":26059},[195],[19,26061,241],{"className":26062},[240],[19,26064,26066],{"className":26065},[213],[19,26067,26069],{"className":26068,"style":1111},[217],[19,26070],{},[19,26072],{"className":26073},[674,675],[19,26075],{"className":26076,"style":487},[254],[19,26078,1656],{"className":26079},[468],[19,26081],{"className":26082,"style":487},[254],[19,26084,26086,26089,26092,26095,26098],{"className":26085},[186],[19,26087],{"className":26088,"style":5312},[190],[19,26090,717],{"className":26091},[277],[19,26093,140],{"className":26094},[195],[19,26096,107],{"className":26097},[195],[19,26099,26101,26104],{"className":26100},[674],[19,26102,741],{"className":26103},[674],[19,26105,26107],{"className":26106},[204],[19,26108,26110],{"className":26109},[208],[19,26111,26113],{"className":26112},[213],[19,26114,26116],{"className":26115,"style":6427},[217],[19,26117,26118,26121],{"style":6430},[19,26119],{"className":26120,"style":226},[225],[19,26122,26124],{"className":26123},[230,231,232,233],[19,26125,52],{"className":26126},[195,199,233],[10,26128,26129,26130,26204,26205,14263],{},"This can be simplified further by noting that ",[19,26131,26133,26157],{"className":26132,"translate":23},[27],[19,26134,26136],{"className":26135},[31],[33,26137,26138],{"xmlns":35},[38,26139,26140,26154],{},[41,26141,26142,26144,26146,26148],{},[54,26143,717],{"stretchy":79},[54,26145,140],{},[81,26147,107],{},[719,26149,26150,26152],{},[54,26151,741],{"stretchy":79},[47,26153,52],{},[175,26155,26156],{"encoding":177},"(-1)^n",[19,26158,26160],{"className":26159,"ariaHidden":61},[182],[19,26161,26163,26166,26169,26172,26175],{"className":26162},[186],[19,26164],{"className":26165,"style":5312},[190],[19,26167,717],{"className":26168},[277],[19,26170,140],{"className":26171},[195],[19,26173,107],{"className":26174},[195],[19,26176,26178,26181],{"className":26177},[674],[19,26179,741],{"className":26180},[674],[19,26182,26184],{"className":26183},[204],[19,26185,26187],{"className":26186},[208],[19,26188,26190],{"className":26189},[213],[19,26191,26193],{"className":26192,"style":5443},[217],[19,26194,26195,26198],{"style":5446},[19,26196],{"className":26197,"style":226},[225],[19,26199,26201],{"className":26200},[230,231,232,233],[19,26202,52],{"className":26203},[195,199,233]," is equal to ",[19,26206,26208,26270],{"className":26207,"translate":23},[27],[19,26209,26211],{"className":26210},[31],[33,26212,26213],{"xmlns":35},[38,26214,26215,26267],{},[41,26216,26217,26219,26221,26223,26235,26237,26239,26241,26243,26245,26247,26249,26251,26253,26255],{},[54,26218,717],{"stretchy":79},[54,26220,140],{},[81,26222,107],{},[719,26224,26225,26227],{},[54,26226,741],{"stretchy":79},[41,26228,26229,26231,26233],{},[47,26230,52],{},[54,26232,140],{},[81,26234,107],{},[54,26236,1656],{},[54,26238,717],{"stretchy":79},[54,26240,140],{},[81,26242,107],{},[54,26244,741],{"stretchy":79},[54,26246,56],{},[54,26248,140],{},[54,26250,717],{"stretchy":79},[54,26252,140],{},[81,26254,107],{},[719,26256,26257,26259],{},[54,26258,741],{"stretchy":79},[41,26260,26261,26263,26265],{},[47,26262,52],{},[54,26264,140],{},[81,26266,107],{},[175,26268,26269],{"encoding":177},"(-1)^{n-1} \\cdot (-1) = -(-1)^{n-1}",[19,26271,26273,26337,26364],{"className":26272,"ariaHidden":61},[182],[19,26274,26276,26280,26283,26286,26289,26328,26331,26334],{"className":26275},[186],[19,26277],{"className":26278,"style":26279},[190],"height:1.0641em;vertical-align:-0.25em;",[19,26281,717],{"className":26282},[277],[19,26284,140],{"className":26285},[195],[19,26287,107],{"className":26288},[195],[19,26290,26292,26295],{"className":26291},[674],[19,26293,741],{"className":26294},[674],[19,26296,26298],{"className":26297},[204],[19,26299,26301],{"className":26300},[208],[19,26302,26304],{"className":26303},[213],[19,26305,26308],{"className":26306,"style":26307},[217],"height:0.8141em;",[19,26309,26310,26313],{"style":5446},[19,26311],{"className":26312,"style":226},[225],[19,26314,26316],{"className":26315},[230,231,232,233],[19,26317,26319,26322,26325],{"className":26318},[195,233],[19,26320,52],{"className":26321},[195,199,233],[19,26323,140],{"className":26324},[468,233],[19,26326,107],{"className":26327},[195,233],[19,26329],{"className":26330,"style":487},[254],[19,26332,1656],{"className":26333},[468],[19,26335],{"className":26336,"style":487},[254],[19,26338,26340,26343,26346,26349,26352,26355,26358,26361],{"className":26339},[186],[19,26341],{"className":26342,"style":5312},[190],[19,26344,717],{"className":26345},[277],[19,26347,140],{"className":26348},[195],[19,26350,107],{"className":26351},[195],[19,26353,741],{"className":26354},[674],[19,26356],{"className":26357,"style":255},[254],[19,26359,56],{"className":26360},[259],[19,26362],{"className":26363,"style":255},[254],[19,26365,26367,26370,26373,26376,26379,26382],{"className":26366},[186],[19,26368],{"className":26369,"style":26279},[190],[19,26371,140],{"className":26372},[195],[19,26374,717],{"className":26375},[277],[19,26377,140],{"className":26378},[195],[19,26380,107],{"className":26381},[195],[19,26383,26385,26388],{"className":26384},[674],[19,26386,741],{"className":26387},[674],[19,26389,26391],{"className":26390},[204],[19,26392,26394],{"className":26393},[208],[19,26395,26397],{"className":26396},[213],[19,26398,26400],{"className":26399,"style":26307},[217],[19,26401,26402,26405],{"style":5446},[19,26403],{"className":26404,"style":226},[225],[19,26406,26408],{"className":26407},[230,231,232,233],[19,26409,26411,26414,26417],{"className":26410},[195,233],[19,26412,52],{"className":26413},[195,199,233],[19,26415,140],{"className":26416},[468,233],[19,26418,107],{"className":26419},[195,233],[19,26421,26423],{"className":26422,"translate":23},[22],[19,26424,26426,26480],{"className":26425,"translate":23},[27],[19,26427,26429],{"className":26428},[31],[33,26430,26431],{"xmlns":35,"display":36},[38,26432,26433,26477],{},[41,26434,26435,26441,26443,26449,26451,26457,26459,26461,26463,26465],{},[44,26436,26437,26439],{},[47,26438,49],{},[47,26440,52],{},[54,26442,56],{},[703,26444,26445,26447],{},[81,26446,107],{},[81,26448,158],{},[54,26450,140],{},[703,26452,26453,26455],{},[81,26454,107],{},[81,26456,158],{},[54,26458,1656],{},[54,26460,717],{"stretchy":79},[54,26462,140],{},[81,26464,107],{},[719,26466,26467,26469],{},[54,26468,741],{"stretchy":79},[41,26470,26471,26473,26475],{},[47,26472,52],{},[54,26474,140],{},[81,26476,107],{},[175,26478,26479],{"encoding":177},"F_n = \\frac{1}{2} - \\frac{1}{2} \\cdot 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we have successfully derived the same result using the difference equation method.",[10,27356,27357],{},"In summary, both the matrix method and the difference equation method lead to the same closed-form expression for the Fibonacci sequence, demonstrating the beauty of mathematics in providing multiple ways to arrive at a solution.",{"title":27359,"searchDepth":27360,"depth":27360,"links":27361},"",2,[27362,27363],{"id":1328,"depth":27360,"text":1329},{"id":20423,"depth":27360,"text":20424},"In Fibonacci's work The Book of Calculation the Fibonacci sequence is defined as follows:",false,true,"md",null,{},"2023-11-20","\u002Fblog\u002F2023\u002F2023-11-20-conduct-the-fibonacci",{"title":5,"description":27364},"blog\u002F2023\u002F2023-11-20-conduct-the-fibonacci","This article gives two methods to derive Fibonacci sequence: matrix method and difference equation method",[27376],"mathematics","E4NIORLoh4P1NyzOwinhwiXVtnk38bnnCIimToDaQvU",1777090298951]