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",[574,1148,1150,1173],{"className":1149},[577],[574,1151,1153],{"className":1152},[581],[583,1154,1155],{"xmlns":585},[587,1156,1157,1171],{},[590,1158,1159,1161,1163,1169],{},[596,1160,705],{},[707,1162,710],{"stretchy":709},[593,1164,1165,1167],{},[596,1166,598],{},[600,1168,602],{},[707,1170,719],{"stretchy":709},[604,1172,722],{"encoding":606},[574,1174,1176],{"className":1175,"ariaHidden":612},[611],[574,1177,1179,1182,1185,1188,1228],{"className":1178},[616],[574,1180],{"className":1181,"style":732},[620],[574,1183,705],{"className":1184},[625,629],[574,1186,710],{"className":1187},[739],[574,1189,1191,1194],{"className":1190},[625],[574,1192,598],{"className":1193},[625,629],[574,1195,1197],{"className":1196},[633],[574,1198,1200,1220],{"className":1199},[637,638],[574,1201,1203,1217],{"className":1202},[642],[574,1204,1206],{"className":1205,"style":647},[646],[574,1207,1208,1211],{"style":650},[574,1209],{"className":1210,"style":655},[654],[574,1212,1214],{"className":1213},[659,660,661,662],[574,1215,602],{"className":1216},[625,662],[574,1218,670],{"className":1219},[669],[574,1221,1223],{"className":1222},[642],[574,1224,1226],{"className":1225,"style":677},[646],[574,1227],{},[574,1229,719],{"className":1230},[783],", der Schritt liefert daraus ",[574,1233,1235,1263],{"className":1234},[577],[574,1236,1238],{"className":1237},[581],[583,1239,1240],{"xmlns":585},[587,1241,1242,1260],{},[590,1243,1244,1246,1248,1254,1256,1258],{},[596,1245,705],{},[707,1247,710],{"stretchy":709},[593,1249,1250,1252],{},[596,1251,598],{},[600,1253,602],{},[707,1255,957],{},[600,1257,960],{},[707,1259,719],{"stretchy":709},[604,1261,1262],{"encoding":606},"A(n_0 + 1)",[574,1264,1266,1327],{"className":1265,"ariaHidden":612},[611],[574,1267,1269,1272,1275,1278,1318,1321,1324],{"className":1268},[616],[574,1270],{"className":1271,"style":732},[620],[574,1273,705],{"className":1274},[625,629],[574,1276,710],{"className":1277},[739],[574,1279,1281,1284],{"className":1280},[625],[574,1282,598],{"className":1283},[625,629],[574,1285,1287],{"className":1286},[633],[574,1288,1290,1310],{"className":1289},[637,638],[574,1291,1293,1307],{"className":1292},[642],[574,1294,1296],{"className":1295,"style":647},[646],[574,1297,1298,1301],{"style":650},[574,1299],{"className":1300,"style":655},[654],[574,1302,1304],{"className":1303},[659,660,661,662],[574,1305,602],{"className":1306},[625,662],[574,1308,670],{"className":1309},[669],[574,1311,1313],{"className":1312},[642],[574,1314,1316],{"className":1315,"style":677},[646],[574,1317],{},[574,1319],{"className":1320,"style":987},[833],[574,1322,957],{"className":1323},[991],[574,1325],{"className":1326,"style":987},[833],[574,1328,1330,1333,1336],{"className":1329},[616],[574,1331],{"className":1332,"style":732},[620],[574,1334,960],{"className":1335},[625],[574,1337,719],{"className":1338},[783],", daraus ",[574,1341,1343,1372],{"className":1342},[577],[574,1344,1346],{"className":1345},[581],[583,1347,1348],{"xmlns":585},[587,1349,1350,1369],{},[590,1351,1352,1354,1356,1362,1364,1367],{},[596,1353,705],{},[707,1355,710],{"stretchy":709},[593,1357,1358,1360],{},[596,1359,598],{},[600,1361,602],{},[707,1363,957],{},[600,1365,1366],{},"2",[707,1368,719],{"stretchy":709},[604,1370,1371],{"encoding":606},"A(n_0 + 2)",[574,1373,1375,1436],{"className":1374,"ariaHidden":612},[611],[574,1376,1378,1381,1384,1387,1427,1430,1433],{"className":1377},[616],[574,1379],{"className":1380,"style":732},[620],[574,1382,705],{"className":1383},[625,629],[574,1385,710],{"className":1386},[739],[574,1388,1390,1393],{"className":1389},[625],[574,1391,598],{"className":1392},[625,629],[574,1394,1396],{"className":1395},[633],[574,1397,1399,1419],{"className":1398},[637,638],[574,1400,1402,1416],{"className":1401},[642],[574,1403,1405],{"className":1404,"style":647},[646],[574,1406,1407,1410],{"style":650},[574,1408],{"className":1409,"style":655},[654],[574,1411,1413],{"className":1412},[659,660,661,662],[574,1414,602],{"className":1415},[625,662],[574,1417,670],{"className":1418},[669],[574,1420,1422],{"className":1421},[642],[574,1423,1425],{"className":1424,"style":677},[646],[574,1426],{},[574,1428],{"className":1429,"style":987},[833],[574,1431,957],{"className":1432},[991],[574,1434],{"className":1435,"style":987},[833],[574,1437,1439,1442,1445],{"className":1438},[616],[574,1440],{"className":1441,"style":732},[620],[574,1443,1366],{"className":1444},[625],[574,1446,719],{"className":1447},[783]," und so weiter. Begründet wird das durch das Induktionsaxiom der natürlichen Zahlen, gleichwertig durch die Wohlordnung: Jede nichtleere Menge natürlicher Zahlen hat ein kleinstes Element, also kann es kein kleinstes Gegenbeispiel geben.",[565,1450,1451,1452,1455,1456,1590],{},"Eine Variante ist die ",[569,1453,1454],{},"starke Induktion",": Im Schritt darf man ",[574,1457,1459,1499],{"className":1458},[577],[574,1460,1462],{"className":1461},[581],[583,1463,1464],{"xmlns":585},[587,1465,1466,1496],{},[590,1467,1468,1470,1472,1478,1480,1483,1486,1488,1490,1492,1494],{},[596,1469,705],{},[707,1471,710],{"stretchy":709},[593,1473,1474,1476],{},[596,1475,598],{},[600,1477,602],{},[707,1479,719],{"stretchy":709},[707,1481,1482],{"separator":612},",",[707,1484,1485],{},"…",[707,1487,1482],{"separator":612},[596,1489,705],{},[707,1491,710],{"stretchy":709},[596,1493,598],{},[707,1495,719],{"stretchy":709},[604,1497,1498],{"encoding":606},"A(n_0), \\ldots, A(n)",[574,1500,1502],{"className":1501,"ariaHidden":612},[611],[574,1503,1505,1508,1511,1514,1554,1557,1561,1565,1569,1572,1575,1578,1581,1584,1587],{"className":1504},[616],[574,1506],{"className":1507,"style":732},[620],[574,1509,705],{"className":1510},[625,629],[574,1512,710],{"className":1513},[739],[574,1515,1517,1520],{"className":1516},[625],[574,1518,598],{"className":1519},[625,629],[574,1521,1523],{"className":1522},[633],[574,1524,1526,1546],{"className":1525},[637,638],[574,1527,1529,1543],{"className":1528},[642],[574,1530,1532],{"className":1531,"style":647},[646],[574,1533,1534,1537],{"style":650},[574,1535],{"className":1536,"style":655},[654],[574,1538,1540],{"className":1539},[659,660,661,662],[574,1541,602],{"className":1542},[625,662],[574,1544,670],{"className":1545},[669],[574,1547,1549],{"className":1548},[642],[574,1550,1552],{"className":1551,"style":677},[646],[574,1553],{},[574,1555,719],{"className":1556},[783],[574,1558,1482],{"className":1559},[1560],"mpunct",[574,1562],{"className":1563,"style":1564},[833],"margin-right:0.1667em;",[574,1566,1485],{"className":1567},[1568],"minner",[574,1570],{"className":1571,"style":1564},[833],[574,1573,1482],{"className":1574},[1560],[574,1576],{"className":1577,"style":1564},[833],[574,1579,705],{"className":1580},[625,629],[574,1582,710],{"className":1583},[739],[574,1585,598],{"className":1586},[625,629],[574,1588,719],{"className":1589},[783]," gleichzeitig voraussetzen. Das braucht man etwa bei Rekursionen, die auf zwei Vorgänger zurückgreifen.",[560,1592,1594],{"id":1593},"beispiele","Beispiele",[565,1596,1597,1600,1601,1793,1794,1007],{},[569,1598,1599],{},"Summe der ungeraden Zahlen:"," Behauptung ",[574,1602,1604,1652],{"className":1603},[577],[574,1605,1607],{"className":1606},[581],[583,1608,1609],{"xmlns":585},[587,1610,1611,1649],{},[590,1612,1613,1615,1617,1620,1622,1624,1626,1628,1630,1632,1635,1637,1639,1642],{},[600,1614,960],{},[707,1616,957],{},[600,1618,1619],{},"3",[707,1621,957],{},[707,1623,1485],{},[707,1625,957],{},[707,1627,710],{"stretchy":709},[600,1629,1366],{},[596,1631,598],{},[707,1633,1634],{},"−",[600,1636,960],{},[707,1638,719],{"stretchy":709},[707,1640,1641],{},"=",[1643,1644,1645,1647],"msup",{},[596,1646,598],{},[600,1648,1366],{},[604,1650,1651],{"encoding":606},"1 + 3 + \\ldots + (2n-1) = n^2",[574,1653,1655,1674,1692,1711,1735,1756],{"className":1654,"ariaHidden":612},[611],[574,1656,1658,1662,1665,1668,1671],{"className":1657},[616],[574,1659],{"className":1660,"style":1661},[620],"height:0.7278em;vertical-align:-0.0833em;",[574,1663,960],{"className":1664},[625],[574,1666],{"className":1667,"style":987},[833],[574,1669,957],{"className":1670},[991],[574,1672],{"className":1673,"style":987},[833],[574,1675,1677,1680,1683,1686,1689],{"className":1676},[616],[574,1678],{"className":1679,"style":1661},[620],[574,1681,1619],{"className":1682},[625],[574,1684],{"className":1685,"style":987},[833],[574,1687,957],{"className":1688},[991],[574,1690],{"className":1691,"style":987},[833],[574,1693,1695,1699,1702,1705,1708],{"className":1694},[616],[574,1696],{"className":1697,"style":1698},[620],"height:0.6667em;vertical-align:-0.0833em;",[574,1700,1485],{"className":1701},[1568],[574,1703],{"className":1704,"style":987},[833],[574,1706,957],{"className":1707},[991],[574,1709],{"className":1710,"style":987},[833],[574,1712,1714,1717,1720,1723,1726,1729,1732],{"className":1713},[616],[574,1715],{"className":1716,"style":732},[620],[574,1718,710],{"className":1719},[739],[574,1721,1366],{"className":1722},[625],[574,1724,598],{"className":1725},[625,629],[574,1727],{"className":1728,"style":987},[833],[574,1730,1634],{"className":1731},[991],[574,1733],{"className":1734,"style":987},[833],[574,1736,1738,1741,1744,1747,1750,1753],{"className":1737},[616],[574,1739],{"className":1740,"style":732},[620],[574,1742,960],{"className":1743},[625],[574,1745,719],{"className":1746},[783],[574,1748],{"className":1749,"style":834},[833],[574,1751,1641],{"className":1752},[838],[574,1754],{"className":1755,"style":834},[833],[574,1757,1759,1763],{"className":1758},[616],[574,1760],{"className":1761,"style":1762},[620],"height:0.8141em;",[574,1764,1766,1769],{"className":1765},[625],[574,1767,598],{"className":1768},[625,629],[574,1770,1772],{"className":1771},[633],[574,1773,1775],{"className":1774},[637],[574,1776,1778],{"className":1777},[642],[574,1779,1781],{"className":1780,"style":1762},[646],[574,1782,1784,1787],{"style":1783},"top:-3.063em;margin-right:0.05em;",[574,1785],{"className":1786,"style":655},[654],[574,1788,1790],{"className":1789},[659,660,661,662],[574,1791,1366],{"className":1792},[625,662]," für ",[574,1795,1797,1815],{"className":1796},[577],[574,1798,1800],{"className":1799},[581],[583,1801,1802],{"xmlns":585},[587,1803,1804,1812],{},[590,1805,1806,1808,1810],{},[596,1807,598],{},[707,1809,807],{},[600,1811,960],{},[604,1813,1814],{"encoding":606},"n \\geq 1",[574,1816,1818,1836],{"className":1817,"ariaHidden":612},[611],[574,1819,1821,1824,1827,1830,1833],{"className":1820},[616],[574,1822],{"className":1823,"style":826},[620],[574,1825,598],{"className":1826},[625,629],[574,1828],{"className":1829,"style":834},[833],[574,1831,807],{"className":1832},[838],[574,1834],{"className":1835,"style":834},[833],[574,1837,1839,1843],{"className":1838},[616],[574,1840],{"className":1841,"style":1842},[620],"height:0.6444em;",[574,1844,960],{"className":1845},[625],[1847,1848,1849,2015],"ul",{},[685,1850,1851,1852,1904,1905,1933,1934,1007],{},"Anfang ",[574,1853,1855,1873],{"className":1854},[577],[574,1856,1858],{"className":1857},[581],[583,1859,1860],{"xmlns":585},[587,1861,1862,1870],{},[590,1863,1864,1866,1868],{},[596,1865,598],{},[707,1867,1641],{},[600,1869,960],{},[604,1871,1872],{"encoding":606},"n = 1",[574,1874,1876,1895],{"className":1875,"ariaHidden":612},[611],[574,1877,1879,1883,1886,1889,1892],{"className":1878},[616],[574,1880],{"className":1881,"style":1882},[620],"height:0.4306em;",[574,1884,598],{"className":1885},[625,629],[574,1887],{"className":1888,"style":834},[833],[574,1890,1641],{"className":1891},[838],[574,1893],{"className":1894,"style":834},[833],[574,1896,1898,1901],{"className":1897},[616],[574,1899],{"className":1900,"style":1842},[620],[574,1902,960],{"className":1903},[625],": links steht ",[574,1906,1908,1921],{"className":1907},[577],[574,1909,1911],{"className":1910},[581],[583,1912,1913],{"xmlns":585},[587,1914,1915,1919],{},[590,1916,1917],{},[600,1918,960],{},[604,1920,960],{"encoding":606},[574,1922,1924],{"className":1923,"ariaHidden":612},[611],[574,1925,1927,1930],{"className":1926},[616],[574,1928],{"className":1929,"style":1842},[620],[574,1931,960],{"className":1932},[625],", rechts ",[574,1935,1937,1959],{"className":1936},[577],[574,1938,1940],{"className":1939},[581],[583,1941,1942],{"xmlns":585},[587,1943,1944,1956],{},[590,1945,1946,1952,1954],{},[1643,1947,1948,1950],{},[600,1949,960],{},[600,1951,1366],{},[707,1953,1641],{},[600,1955,960],{},[604,1957,1958],{"encoding":606},"1^2 = 1",[574,1960,1962,2006],{"className":1961,"ariaHidden":612},[611],[574,1963,1965,1968,1997,2000,2003],{"className":1964},[616],[574,1966],{"className":1967,"style":1762},[620],[574,1969,1971,1974],{"className":1970},[625],[574,1972,960],{"className":1973},[625],[574,1975,1977],{"className":1976},[633],[574,1978,1980],{"className":1979},[637],[574,1981,1983],{"className":1982},[642],[574,1984,1986],{"className":1985,"style":1762},[646],[574,1987,1988,1991],{"style":1783},[574,1989],{"className":1990,"style":655},[654],[574,1992,1994],{"className":1993},[659,660,661,662],[574,1995,1366],{"className":1996},[625,662],[574,1998],{"className":1999,"style":834},[833],[574,2001,1641],{"className":2002},[838],[574,2004],{"className":2005,"style":834},[833],[574,2007,2009,2012],{"className":2008},[616],[574,2010],{"className":2011,"style":1842},[620],[574,2013,960],{"className":2014},[625],[685,2016,2017,2018,2046],{},"Schritt: Gilt die Formel für ",[574,2019,2021,2034],{"className":2020},[577],[574,2022,2024],{"className":2023},[581],[583,2025,2026],{"xmlns":585},[587,2027,2028,2032],{},[590,2029,2030],{},[596,2031,598],{},[604,2033,598],{"encoding":606},[574,2035,2037],{"className":2036,"ariaHidden":612},[611],[574,2038,2040,2043],{"className":2039},[616],[574,2041],{"className":2042,"style":1882},[620],[574,2044,598],{"className":2045},[625,629],", so ist",[574,2048,2051],{"className":2049},[2050],"katex-display",[574,2052,2054,2147],{"className":2053},[577],[574,2055,2057],{"className":2056},[581],[583,2058,2060],{"xmlns":585,"display":2059},"block",[587,2061,2062,2144],{},[590,2063,2064,2066,2068,2070,2072,2074,2076,2078,2080,2082,2084,2086,2088,2090,2092,2094,2096,2098,2100,2102,2104,2106,2108,2110,2112,2118,2120,2122,2124,2126,2128,2130,2132,2134,2136,2138],{},[600,2065,960],{},[707,2067,957],{},[600,2069,1619],{},[707,2071,957],{},[707,2073,1485],{},[707,2075,957],{},[707,2077,710],{"stretchy":709},[600,2079,1366],{},[596,2081,598],{},[707,2083,1634],{},[600,2085,960],{},[707,2087,719],{"stretchy":709},[707,2089,957],{},[707,2091,710],{"stretchy":709},[600,2093,1366],{},[707,2095,710],{"stretchy":709},[596,2097,598],{},[707,2099,957],{},[600,2101,960],{},[707,2103,719],{"stretchy":709},[707,2105,1634],{},[600,2107,960],{},[707,2109,719],{"stretchy":709},[707,2111,1641],{},[1643,2113,2114,2116],{},[596,2115,598],{},[600,2117,1366],{},[707,2119,957],{},[600,2121,1366],{},[596,2123,598],{},[707,2125,957],{},[600,2127,960],{},[707,2129,1641],{},[707,2131,710],{"stretchy":709},[596,2133,598],{},[707,2135,957],{},[600,2137,960],{},[1643,2139,2140,2142],{},[707,2141,719],{"stretchy":709},[600,2143,1366],{},[604,2145,2146],{"encoding":606},"1 + 3 + \\ldots + (2n-1) + (2(n+1) - 1) = n^2 + 2n + 1 = (n+1)^2",[574,2148,2150,2168,2186,2204,2228,2249,2276,2297,2318,2365,2386,2404,2425],{"className":2149,"ariaHidden":612},[611],[574,2151,2153,2156,2159,2162,2165],{"className":2152},[616],[574,2154],{"className":2155,"style":1661},[620],[574,2157,960],{"className":2158},[625],[574,2160],{"className":2161,"style":987},[833],[574,2163,957],{"className":2164},[991],[574,2166],{"className":2167,"style":987},[833],[574,2169,2171,2174,2177,2180,2183],{"className":2170},[616],[574,2172],{"className":2173,"style":1661},[620],[574,2175,1619],{"className":2176},[625],[574,2178],{"className":2179,"style":987},[833],[574,2181,957],{"className":2182},[991],[574,2184],{"className":2185,"style":987},[833],[574,2187,2189,2192,2195,2198,2201],{"className":2188},[616],[574,2190],{"className":2191,"style":1698},[620],[574,2193,1485],{"className":2194},[1568],[574,2196],{"className":2197,"style":987},[833],[574,2199,957],{"className":2200},[991],[574,2202],{"className":2203,"style":987},[833],[574,2205,2207,2210,2213,2216,2219,2222,2225],{"className":2206},[616],[574,2208],{"className":2209,"style":732},[620],[574,2211,710],{"className":2212},[739],[574,2214,1366],{"className":2215},[625],[574,2217,598],{"className":2218},[625,629],[574,2220],{"className":2221,"style":987},[833],[574,2223,1634],{"className":2224},[991],[574,2226],{"className":2227,"style":987},[833],[574,2229,2231,2234,2237,2240,2243,2246],{"className":2230},[616],[574,2232],{"className":2233,"style":732},[620],[574,2235,960],{"className":2236},[625],[574,2238,719],{"className":2239},[783],[574,2241],{"className":2242,"style":987},[833],[574,2244,957],{"className":2245},[991],[574,2247],{"className":2248,"style":987},[833],[574,2250,2252,2255,2258,2261,2264,2267,2270,2273],{"className":2251},[616],[574,2253],{"className":2254,"style":732},[620],[574,2256,710],{"className":2257},[739],[574,2259,1366],{"className":2260},[625],[574,2262,710],{"className":2263},[739],[574,2265,598],{"className":2266},[625,629],[574,2268],{"className":2269,"style":987},[833],[574,2271,957],{"className":2272},[991],[574,2274],{"className":2275,"style":987},[833],[574,2277,2279,2282,2285,2288,2291,2294],{"className":2278},[616],[574,2280],{"className":2281,"style":732},[620],[574,2283,960],{"className":2284},[625],[574,2286,719],{"className":2287},[783],[574,2289],{"className":2290,"style":987},[833],[574,2292,1634],{"className":2293},[991],[574,2295],{"className":2296,"style":987},[833],[574,2298,2300,2303,2306,2309,2312,2315],{"className":2299},[616],[574,2301],{"className":2302,"style":732},[620],[574,2304,960],{"className":2305},[625],[574,2307,719],{"className":2308},[783],[574,2310],{"className":2311,"style":834},[833],[574,2313,1641],{"className":2314},[838],[574,2316],{"className":2317,"style":834},[833],[574,2319,2321,2325,2356,2359,2362],{"className":2320},[616],[574,2322],{"className":2323,"style":2324},[620],"height:0.9474em;vertical-align:-0.0833em;",[574,2326,2328,2331],{"className":2327},[625],[574,2329,598],{"className":2330},[625,629],[574,2332,2334],{"className":2333},[633],[574,2335,2337],{"className":2336},[637],[574,2338,2340],{"className":2339},[642],[574,2341,2344],{"className":2342,"style":2343},[646],"height:0.8641em;",[574,2345,2347,2350],{"style":2346},"top:-3.113em;margin-right:0.05em;",[574,2348],{"className":2349,"style":655},[654],[574,2351,2353],{"className":2352},[659,660,661,662],[574,2354,1366],{"className":2355},[625,662],[574,2357],{"className":2358,"style":987},[833],[574,2360,957],{"className":2361},[991],[574,2363],{"className":2364,"style":987},[833],[574,2366,2368,2371,2374,2377,2380,2383],{"className":2367},[616],[574,2369],{"className":2370,"style":1661},[620],[574,2372,1366],{"className":2373},[625],[574,2375,598],{"className":2376},[625,629],[574,2378],{"className":2379,"style":987},[833],[574,2381,957],{"className":2382},[991],[574,2384],{"className":2385,"style":987},[833],[574,2387,2389,2392,2395,2398,2401],{"className":2388},[616],[574,2390],{"className":2391,"style":1842},[620],[574,2393,960],{"className":2394},[625],[574,2396],{"className":2397,"style":834},[833],[574,2399,1641],{"className":2400},[838],[574,2402],{"className":2403,"style":834},[833],[574,2405,2407,2410,2413,2416,2419,2422],{"className":2406},[616],[574,2408],{"className":2409,"style":732},[620],[574,2411,710],{"className":2412},[739],[574,2414,598],{"className":2415},[625,629],[574,2417],{"className":2418,"style":987},[833],[574,2420,957],{"className":2421},[991],[574,2423],{"className":2424,"style":987},[833],[574,2426,2428,2432,2435],{"className":2427},[616],[574,2429],{"className":2430,"style":2431},[620],"height:1.1141em;vertical-align:-0.25em;",[574,2433,960],{"className":2434},[625],[574,2436,2438,2441],{"className":2437},[783],[574,2439,719],{"className":2440},[783],[574,2442,2444],{"className":2443},[633],[574,2445,2447],{"className":2446},[637],[574,2448,2450],{"className":2449},[642],[574,2451,2453],{"className":2452,"style":2343},[646],[574,2454,2455,2458],{"style":2346},[574,2456],{"className":2457,"style":655},[654],[574,2459,2461],{"className":2460},[659,660,661,662],[574,2462,1366],{"className":2463},[625,662],[565,2465,2466,2467,2519,2520,1007],{},"Probe für ",[574,2468,2470,2489],{"className":2469},[577],[574,2471,2473],{"className":2472},[581],[583,2474,2475],{"xmlns":585},[587,2476,2477,2486],{},[590,2478,2479,2481,2483],{},[596,2480,598],{},[707,2482,1641],{},[600,2484,2485],{},"5",[604,2487,2488],{"encoding":606},"n = 5",[574,2490,2492,2510],{"className":2491,"ariaHidden":612},[611],[574,2493,2495,2498,2501,2504,2507],{"className":2494},[616],[574,2496],{"className":2497,"style":1882},[620],[574,2499,598],{"className":2500},[625,629],[574,2502],{"className":2503,"style":834},[833],[574,2505,1641],{"className":2506},[838],[574,2508],{"className":2509,"style":834},[833],[574,2511,2513,2516],{"className":2512},[616],[574,2514],{"className":2515,"style":1842},[620],[574,2517,2485],{"className":2518},[625],": ",[574,2521,2523,2568],{"className":2522},[577],[574,2524,2526],{"className":2525},[581],[583,2527,2528],{"xmlns":585},[587,2529,2530,2565],{},[590,2531,2532,2534,2536,2538,2540,2542,2544,2547,2549,2552,2554,2557,2559],{},[600,2533,960],{},[707,2535,957],{},[600,2537,1619],{},[707,2539,957],{},[600,2541,2485],{},[707,2543,957],{},[600,2545,2546],{},"7",[707,2548,957],{},[600,2550,2551],{},"9",[707,2553,1641],{},[600,2555,2556],{},"25",[707,2558,1641],{},[1643,2560,2561,2563],{},[600,2562,2485],{},[600,2564,1366],{},[604,2566,2567],{"encoding":606},"1 + 3 + 5 + 7 + 9 = 25 = 5^2",[574,2569,2571,2589,2607,2625,2643,2661,2679],{"className":2570,"ariaHidden":612},[611],[574,2572,2574,2577,2580,2583,2586],{"className":2573},[616],[574,2575],{"className":2576,"style":1661},[620],[574,2578,960],{"className":2579},[625],[574,2581],{"className":2582,"style":987},[833],[574,2584,957],{"className":2585},[991],[574,2587],{"className":2588,"style":987},[833],[574,2590,2592,2595,2598,2601,2604],{"className":2591},[616],[574,2593],{"className":2594,"style":1661},[620],[574,2596,1619],{"className":2597},[625],[574,2599],{"className":2600,"style":987},[833],[574,2602,957],{"className":2603},[991],[574,2605],{"className":2606,"style":987},[833],[574,2608,2610,2613,2616,2619,2622],{"className":2609},[616],[574,2611],{"className":2612,"style":1661},[620],[574,2614,2485],{"className":2615},[625],[574,2617],{"className":2618,"style":987},[833],[574,2620,957],{"className":2621},[991],[574,2623],{"className":2624,"style":987},[833],[574,2626,2628,2631,2634,2637,2640],{"className":2627},[616],[574,2629],{"className":2630,"style":1661},[620],[574,2632,2546],{"className":2633},[625],[574,2635],{"className":2636,"style":987},[833],[574,2638,957],{"className":2639},[991],[574,2641],{"className":2642,"style":987},[833],[574,2644,2646,2649,2652,2655,2658],{"className":2645},[616],[574,2647],{"className":2648,"style":1842},[620],[574,2650,2551],{"className":2651},[625],[574,2653],{"className":2654,"style":834},[833],[574,2656,1641],{"className":2657},[838],[574,2659],{"className":2660,"style":834},[833],[574,2662,2664,2667,2670,2673,2676],{"className":2663},[616],[574,2665],{"className":2666,"style":1842},[620],[574,2668,2556],{"className":2669},[625],[574,2671],{"className":2672,"style":834},[833],[574,2674,1641],{"className":2675},[838],[574,2677],{"className":2678,"style":834},[833],[574,2680,2682,2685],{"className":2681},[616],[574,2683],{"className":2684,"style":1762},[620],[574,2686,2688,2691],{"className":2687},[625],[574,2689,2485],{"className":2690},[625],[574,2692,2694],{"className":2693},[633],[574,2695,2697],{"className":2696},[637],[574,2698,2700],{"className":2699},[642],[574,2701,2703],{"className":2702,"style":1762},[646],[574,2704,2705,2708],{"style":1783},[574,2706],{"className":2707,"style":655},[654],[574,2709,2711],{"className":2710},[659,660,661,662],[574,2712,1366],{"className":2713},[625,662],[565,2715,2716,1600,2719,1054,2824,1007],{},[569,2717,2718],{},"Teilbarkeit:",[574,2720,2722,2749],{"className":2721},[577],[574,2723,2725],{"className":2724},[581],[583,2726,2727],{"xmlns":585},[587,2728,2729,2746],{},[590,2730,2731,2733,2736,2742,2744],{},[600,2732,1619],{},[707,2734,2735],{},"∣",[1643,2737,2738,2740],{},[596,2739,598],{},[600,2741,1619],{},[707,2743,1634],{},[596,2745,598],{},[604,2747,2748],{"encoding":606},"3 \\mid n^3 - n",[574,2750,2752,2770,2815],{"className":2751,"ariaHidden":612},[611],[574,2753,2755,2758,2761,2764,2767],{"className":2754},[616],[574,2756],{"className":2757,"style":732},[620],[574,2759,1619],{"className":2760},[625],[574,2762],{"className":2763,"style":834},[833],[574,2765,2735],{"className":2766},[838],[574,2768],{"className":2769,"style":834},[833],[574,2771,2773,2777,2806,2809,2812],{"className":2772},[616],[574,2774],{"className":2775,"style":2776},[620],"height:0.8974em;vertical-align:-0.0833em;",[574,2778,2780,2783],{"className":2779},[625],[574,2781,598],{"className":2782},[625,629],[574,2784,2786],{"className":2785},[633],[574,2787,2789],{"className":2788},[637],[574,2790,2792],{"className":2791},[642],[574,2793,2795],{"className":2794,"style":1762},[646],[574,2796,2797,2800],{"style":1783},[574,2798],{"className":2799,"style":655},[654],[574,2801,2803],{"className":2802},[659,660,661,662],[574,2804,1619],{"className":2805},[625,662],[574,2807],{"className":2808,"style":987},[833],[574,2810,1634],{"className":2811},[991],[574,2813],{"className":2814,"style":987},[833],[574,2816,2818,2821],{"className":2817},[616],[574,2819],{"className":2820,"style":1882},[620],[574,2822,598],{"className":2823},[625,629],[574,2825,2827,2844],{"className":2826},[577],[574,2828,2830],{"className":2829},[581],[583,2831,2832],{"xmlns":585},[587,2833,2834,2842],{},[590,2835,2836,2838,2840],{},[596,2837,598],{},[707,2839,807],{},[600,2841,960],{},[604,2843,1814],{"encoding":606},[574,2845,2847,2865],{"className":2846,"ariaHidden":612},[611],[574,2848,2850,2853,2856,2859,2862],{"className":2849},[616],[574,2851],{"className":2852,"style":826},[620],[574,2854,598],{"className":2855},[625,629],[574,2857],{"className":2858,"style":834},[833],[574,2860,807],{"className":2861},[838],[574,2863],{"className":2864,"style":834},[833],[574,2866,2868,2871],{"className":2867},[616],[574,2869],{"className":2870,"style":1842},[620],[574,2872,960],{"className":2873},[625],[1847,2875,2876,3031],{},[685,2877,1851,2878,2519,2928,3001,3002,3030],{},[574,2879,2881,2898],{"className":2880},[577],[574,2882,2884],{"className":2883},[581],[583,2885,2886],{"xmlns":585},[587,2887,2888,2896],{},[590,2889,2890,2892,2894],{},[596,2891,598],{},[707,2893,1641],{},[600,2895,960],{},[604,2897,1872],{"encoding":606},[574,2899,2901,2919],{"className":2900,"ariaHidden":612},[611],[574,2902,2904,2907,2910,2913,2916],{"className":2903},[616],[574,2905],{"className":2906,"style":1882},[620],[574,2908,598],{"className":2909},[625,629],[574,2911],{"className":2912,"style":834},[833],[574,2914,1641],{"className":2915},[838],[574,2917],{"className":2918,"style":834},[833],[574,2920,2922,2925],{"className":2921},[616],[574,2923],{"className":2924,"style":1842},[620],[574,2926,960],{"className":2927},[625],[574,2929,2931,2953],{"className":2930},[577],[574,2932,2934],{"className":2933},[581],[583,2935,2936],{"xmlns":585},[587,2937,2938,2950],{},[590,2939,2940,2942,2944,2946,2948],{},[600,2941,960],{},[707,2943,1634],{},[600,2945,960],{},[707,2947,1641],{},[600,2949,602],{},[604,2951,2952],{"encoding":606},"1 - 1 = 0",[574,2954,2956,2974,2992],{"className":2955,"ariaHidden":612},[611],[574,2957,2959,2962,2965,2968,2971],{"className":2958},[616],[574,2960],{"className":2961,"style":1661},[620],[574,2963,960],{"className":2964},[625],[574,2966],{"className":2967,"style":987},[833],[574,2969,1634],{"className":2970},[991],[574,2972],{"className":2973,"style":987},[833],[574,2975,2977,2980,2983,2986,2989],{"className":2976},[616],[574,2978],{"className":2979,"style":1842},[620],[574,2981,960],{"className":2982},[625],[574,2984],{"className":2985,"style":834},[833],[574,2987,1641],{"className":2988},[838],[574,2990],{"className":2991,"style":834},[833],[574,2993,2995,2998],{"className":2994},[616],[574,2996],{"className":2997,"style":1842},[620],[574,2999,602],{"className":3000},[625]," ist durch ",[574,3003,3005,3018],{"className":3004},[577],[574,3006,3008],{"className":3007},[581],[583,3009,3010],{"xmlns":585},[587,3011,3012,3016],{},[590,3013,3014],{},[600,3015,1619],{},[604,3017,1619],{"encoding":606},[574,3019,3021],{"className":3020,"ariaHidden":612},[611],[574,3022,3024,3027],{"className":3023},[616],[574,3025],{"className":3026,"style":1842},[620],[574,3028,1619],{"className":3029},[625]," teilbar.",[685,3032,3033,3034,3353,3354,3382],{},"Schritt: ",[574,3035,3037,3109],{"className":3036},[577],[574,3038,3040],{"className":3039},[581],[583,3041,3042],{"xmlns":585},[587,3043,3044,3106],{},[590,3045,3046,3048,3050,3052,3054,3060,3062,3064,3066,3068,3070,3072,3074,3076,3082,3084,3086,3088,3090,3092,3094,3100,3102,3104],{},[707,3047,710],{"stretchy":709},[596,3049,598],{},[707,3051,957],{},[600,3053,960],{},[1643,3055,3056,3058],{},[707,3057,719],{"stretchy":709},[600,3059,1619],{},[707,3061,1634],{},[707,3063,710],{"stretchy":709},[596,3065,598],{},[707,3067,957],{},[600,3069,960],{},[707,3071,719],{"stretchy":709},[707,3073,1641],{},[707,3075,710],{"stretchy":709},[1643,3077,3078,3080],{},[596,3079,598],{},[600,3081,1619],{},[707,3083,1634],{},[596,3085,598],{},[707,3087,719],{"stretchy":709},[707,3089,957],{},[600,3091,1619],{},[707,3093,710],{"stretchy":709},[1643,3095,3096,3098],{},[596,3097,598],{},[600,3099,1366],{},[707,3101,957],{},[596,3103,598],{},[707,3105,719],{"stretchy":709},[604,3107,3108],{"encoding":606},"(n+1)^3 - (n+1) = (n^3 - n) + 3(n^2 + n)",[574,3110,3112,3133,3181,3202,3223,3270,3291,3341],{"className":3111,"ariaHidden":612},[611],[574,3113,3115,3118,3121,3124,3127,3130],{"className":3114},[616],[574,3116],{"className":3117,"style":732},[620],[574,3119,710],{"className":3120},[739],[574,3122,598],{"className":3123},[625,629],[574,3125],{"className":3126,"style":987},[833],[574,3128,957],{"className":3129},[991],[574,3131],{"className":3132,"style":987},[833],[574,3134,3136,3140,3143,3172,3175,3178],{"className":3135},[616],[574,3137],{"className":3138,"style":3139},[620],"height:1.0641em;vertical-align:-0.25em;",[574,3141,960],{"className":3142},[625],[574,3144,3146,3149],{"className":3145},[783],[574,3147,719],{"className":3148},[783],[574,3150,3152],{"className":3151},[633],[574,3153,3155],{"className":3154},[637],[574,3156,3158],{"className":3157},[642],[574,3159,3161],{"className":3160,"style":1762},[646],[574,3162,3163,3166],{"style":1783},[574,3164],{"className":3165,"style":655},[654],[574,3167,3169],{"className":3168},[659,660,661,662],[574,3170,1619],{"className":3171},[625,662],[574,3173],{"className":3174,"style":987},[833],[574,3176,1634],{"className":3177},[991],[574,3179],{"className":3180,"style":987},[833],[574,3182,3184,3187,3190,3193,3196,3199],{"className":3183},[616],[574,3185],{"className":3186,"style":732},[620],[574,3188,710],{"className":3189},[739],[574,3191,598],{"className":3192},[625,629],[574,3194],{"className":3195,"style":987},[833],[574,3197,957],{"className":3198},[991],[574,3200],{"className":3201,"style":987},[833],[574,3203,3205,3208,3211,3214,3217,3220],{"className":3204},[616],[574,3206],{"className":3207,"style":732},[620],[574,3209,960],{"className":3210},[625],[574,3212,719],{"className":3213},[783],[574,3215],{"className":3216,"style":834},[833],[574,3218,1641],{"className":3219},[838],[574,3221],{"className":3222,"style":834},[833],[574,3224,3226,3229,3232,3261,3264,3267],{"className":3225},[616],[574,3227],{"className":3228,"style":3139},[620],[574,3230,710],{"className":3231},[739],[574,3233,3235,3238],{"className":3234},[625],[574,3236,598],{"className":3237},[625,629],[574,3239,3241],{"className":3240},[633],[574,3242,3244],{"className":3243},[637],[574,3245,3247],{"className":3246},[642],[574,3248,3250],{"className":3249,"style":1762},[646],[574,3251,3252,3255],{"style":1783},[574,3253],{"className":3254,"style":655},[654],[574,3256,3258],{"className":3257},[659,660,661,662],[574,3259,1619],{"className":3260},[625,662],[574,3262],{"className":3263,"style":987},[833],[574,3265,1634],{"className":3266},[991],[574,3268],{"className":3269,"style":987},[833],[574,3271,3273,3276,3279,3282,3285,3288],{"className":3272},[616],[574,3274],{"className":3275,"style":732},[620],[574,3277,598],{"className":3278},[625,629],[574,3280,719],{"className":3281},[783],[574,3283],{"className":3284,"style":987},[833],[574,3286,957],{"className":3287},[991],[574,3289],{"className":3290,"style":987},[833],[574,3292,3294,3297,3300,3303,3332,3335,3338],{"className":3293},[616],[574,3295],{"className":3296,"style":3139},[620],[574,3298,1619],{"className":3299},[625],[574,3301,710],{"className":3302},[739],[574,3304,3306,3309],{"className":3305},[625],[574,3307,598],{"className":3308},[625,629],[574,3310,3312],{"className":3311},[633],[574,3313,3315],{"className":3314},[637],[574,3316,3318],{"className":3317},[642],[574,3319,3321],{"className":3320,"style":1762},[646],[574,3322,3323,3326],{"style":1783},[574,3324],{"className":3325,"style":655},[654],[574,3327,3329],{"className":3328},[659,660,661,662],[574,3330,1366],{"className":3331},[625,662],[574,3333],{"className":3334,"style":987},[833],[574,3336,957],{"className":3337},[991],[574,3339],{"className":3340,"style":987},[833],[574,3342,3344,3347,3350],{"className":3343},[616],[574,3345],{"className":3346,"style":732},[620],[574,3348,598],{"className":3349},[625,629],[574,3351,719],{"className":3352},[783],". Der erste Summand ist nach Voraussetzung durch ",[574,3355,3357,3370],{"className":3356},[577],[574,3358,3360],{"className":3359},[581],[583,3361,3362],{"xmlns":585},[587,3363,3364,3368],{},[590,3365,3366],{},[600,3367,1619],{},[604,3369,1619],{"encoding":606},[574,3371,3373],{"className":3372,"ariaHidden":612},[611],[574,3374,3376,3379],{"className":3375},[616],[574,3377],{"className":3378,"style":1842},[620],[574,3380,1619],{"className":3381},[625]," teilbar, der zweite offensichtlich.",[565,3384,3385,3386,3490,3491,3596,3597,3030],{},"Probe: ",[574,3387,3389,3416],{"className":3388},[577],[574,3390,3392],{"className":3391},[581],[583,3393,3394],{"xmlns":585},[587,3395,3396,3413],{},[590,3397,3398,3404,3406,3408,3410],{},[1643,3399,3400,3402],{},[600,3401,1366],{},[600,3403,1619],{},[707,3405,1634],{},[600,3407,1366],{},[707,3409,1641],{},[600,3411,3412],{},"6",[604,3414,3415],{"encoding":606},"2^3 - 2 = 6",[574,3417,3419,3463,3481],{"className":3418,"ariaHidden":612},[611],[574,3420,3422,3425,3454,3457,3460],{"className":3421},[616],[574,3423],{"className":3424,"style":2776},[620],[574,3426,3428,3431],{"className":3427},[625],[574,3429,1366],{"className":3430},[625],[574,3432,3434],{"className":3433},[633],[574,3435,3437],{"className":3436},[637],[574,3438,3440],{"className":3439},[642],[574,3441,3443],{"className":3442,"style":1762},[646],[574,3444,3445,3448],{"style":1783},[574,3446],{"className":3447,"style":655},[654],[574,3449,3451],{"className":3450},[659,660,661,662],[574,3452,1619],{"className":3453},[625,662],[574,3455],{"className":3456,"style":987},[833],[574,3458,1634],{"className":3459},[991],[574,3461],{"className":3462,"style":987},[833],[574,3464,3466,3469,3472,3475,3478],{"className":3465},[616],[574,3467],{"className":3468,"style":1842},[620],[574,3470,1366],{"className":3471},[625],[574,3473],{"className":3474,"style":834},[833],[574,3476,1641],{"className":3477},[838],[574,3479],{"className":3480,"style":834},[833],[574,3482,3484,3487],{"className":3483},[616],[574,3485],{"className":3486,"style":1842},[620],[574,3488,3412],{"className":3489},[625]," und ",[574,3492,3494,3522],{"className":3493},[577],[574,3495,3497],{"className":3496},[581],[583,3498,3499],{"xmlns":585},[587,3500,3501,3519],{},[590,3502,3503,3510,3512,3514,3516],{},[1643,3504,3505,3508],{},[600,3506,3507],{},"4",[600,3509,1619],{},[707,3511,1634],{},[600,3513,3507],{},[707,3515,1641],{},[600,3517,3518],{},"60",[604,3520,3521],{"encoding":606},"4^3 - 4 = 60",[574,3523,3525,3569,3587],{"className":3524,"ariaHidden":612},[611],[574,3526,3528,3531,3560,3563,3566],{"className":3527},[616],[574,3529],{"className":3530,"style":2776},[620],[574,3532,3534,3537],{"className":3533},[625],[574,3535,3507],{"className":3536},[625],[574,3538,3540],{"className":3539},[633],[574,3541,3543],{"className":3542},[637],[574,3544,3546],{"className":3545},[642],[574,3547,3549],{"className":3548,"style":1762},[646],[574,3550,3551,3554],{"style":1783},[574,3552],{"className":3553,"style":655},[654],[574,3555,3557],{"className":3556},[659,660,661,662],[574,3558,1619],{"className":3559},[625,662],[574,3561],{"className":3562,"style":987},[833],[574,3564,1634],{"className":3565},[991],[574,3567],{"className":3568,"style":987},[833],[574,3570,3572,3575,3578,3581,3584],{"className":3571},[616],[574,3573],{"className":3574,"style":1842},[620],[574,3576,3507],{"className":3577},[625],[574,3579],{"className":3580,"style":834},[833],[574,3582,1641],{"className":3583},[838],[574,3585],{"className":3586,"style":834},[833],[574,3588,3590,3593],{"className":3589},[616],[574,3591],{"className":3592,"style":1842},[620],[574,3594,3518],{"className":3595},[625]," sind durch ",[574,3598,3600,3613],{"className":3599},[577],[574,3601,3603],{"className":3602},[581],[583,3604,3605],{"xmlns":585},[587,3606,3607,3611],{},[590,3608,3609],{},[600,3610,1619],{},[604,3612,1619],{"encoding":606},[574,3614,3616],{"className":3615,"ariaHidden":612},[611],[574,3617,3619,3622],{"className":3618},[616],[574,3620],{"className":3621,"style":1842},[620],[574,3623,1619],{"className":3624},[625],[560,3626,3628],{"id":3627},"typische-fehler","Typische Fehler",[1847,3630,3631,3888,4069,4196],{},[685,3632,3633,3636,3637,3693,3694,3749,3750,3887],{},[569,3634,3635],{},"Induktionsanfang weglassen."," Für die falsche Aussage „",[574,3638,3640,3660],{"className":3639},[577],[574,3641,3643],{"className":3642},[581],[583,3644,3645],{"xmlns":585},[587,3646,3647,3657],{},[590,3648,3649,3651,3653,3655],{},[600,3650,1366],{},[596,3652,598],{},[707,3654,957],{},[600,3656,960],{},[604,3658,3659],{"encoding":606},"2n + 1",[574,3661,3663,3684],{"className":3662,"ariaHidden":612},[611],[574,3664,3666,3669,3672,3675,3678,3681],{"className":3665},[616],[574,3667],{"className":3668,"style":1661},[620],[574,3670,1366],{"className":3671},[625],[574,3673,598],{"className":3674},[625,629],[574,3676],{"className":3677,"style":987},[833],[574,3679,957],{"className":3680},[991],[574,3682],{"className":3683,"style":987},[833],[574,3685,3687,3690],{"className":3686},[616],[574,3688],{"className":3689,"style":1842},[620],[574,3691,960],{"className":3692},[625]," ist gerade“ funktioniert der Schritt: Ist ",[574,3695,3697,3716],{"className":3696},[577],[574,3698,3700],{"className":3699},[581],[583,3701,3702],{"xmlns":585},[587,3703,3704,3714],{},[590,3705,3706,3708,3710,3712],{},[600,3707,1366],{},[596,3709,598],{},[707,3711,957],{},[600,3713,960],{},[604,3715,3659],{"encoding":606},[574,3717,3719,3740],{"className":3718,"ariaHidden":612},[611],[574,3720,3722,3725,3728,3731,3734,3737],{"className":3721},[616],[574,3723],{"className":3724,"style":1661},[620],[574,3726,1366],{"className":3727},[625],[574,3729,598],{"className":3730},[625,629],[574,3732],{"className":3733,"style":987},[833],[574,3735,957],{"className":3736},[991],[574,3738],{"className":3739,"style":987},[833],[574,3741,3743,3746],{"className":3742},[616],[574,3744],{"className":3745,"style":1842},[620],[574,3747,960],{"className":3748},[625]," gerade, dann auch ",[574,3751,3753,3791],{"className":3752},[577],[574,3754,3756],{"className":3755},[581],[583,3757,3758],{"xmlns":585},[587,3759,3760,3788],{},[590,3761,3762,3764,3766,3768,3770,3772,3774,3776,3778,3780,3782,3784,3786],{},[600,3763,1366],{},[596,3765,598],{},[707,3767,957],{},[600,3769,1619],{},[707,3771,1641],{},[707,3773,710],{"stretchy":709},[600,3775,1366],{},[596,3777,598],{},[707,3779,957],{},[600,3781,960],{},[707,3783,719],{"stretchy":709},[707,3785,957],{},[600,3787,1366],{},[604,3789,3790],{"encoding":606},"2n + 3 = (2n+1) + 2",[574,3792,3794,3815,3833,3857,3878],{"className":3793,"ariaHidden":612},[611],[574,3795,3797,3800,3803,3806,3809,3812],{"className":3796},[616],[574,3798],{"className":3799,"style":1661},[620],[574,3801,1366],{"className":3802},[625],[574,3804,598],{"className":3805},[625,629],[574,3807],{"className":3808,"style":987},[833],[574,3810,957],{"className":3811},[991],[574,3813],{"className":3814,"style":987},[833],[574,3816,3818,3821,3824,3827,3830],{"className":3817},[616],[574,3819],{"className":3820,"style":1842},[620],[574,3822,1619],{"className":3823},[625],[574,3825],{"className":3826,"style":834},[833],[574,3828,1641],{"className":3829},[838],[574,3831],{"className":3832,"style":834},[833],[574,3834,3836,3839,3842,3845,3848,3851,3854],{"className":3835},[616],[574,3837],{"className":3838,"style":732},[620],[574,3840,710],{"className":3841},[739],[574,3843,1366],{"className":3844},[625],[574,3846,598],{"className":3847},[625,629],[574,3849],{"className":3850,"style":987},[833],[574,3852,957],{"className":3853},[991],[574,3855],{"className":3856,"style":987},[833],[574,3858,3860,3863,3866,3869,3872,3875],{"className":3859},[616],[574,3861],{"className":3862,"style":732},[620],[574,3864,960],{"className":3865},[625],[574,3867,719],{"className":3868},[783],[574,3870],{"className":3871,"style":987},[833],[574,3873,957],{"className":3874},[991],[574,3876],{"className":3877,"style":987},[833],[574,3879,3881,3884],{"className":3880},[616],[574,3882],{"className":3883,"style":1842},[620],[574,3885,1366],{"className":3886},[625],". Ohne gültigen Anfang beweist das nichts.",[685,3889,3890,3958,3959,4002,4003,4068],{},[569,3891,3892,3957],{},[574,3893,3895,3918],{"className":3894},[577],[574,3896,3898],{"className":3897},[581],[583,3899,3900],{"xmlns":585},[587,3901,3902,3916],{},[590,3903,3904,3906,3908,3910,3912,3914],{},[596,3905,705],{},[707,3907,710],{"stretchy":709},[596,3909,598],{},[707,3911,957],{},[600,3913,960],{},[707,3915,719],{"stretchy":709},[604,3917,965],{"encoding":606},[574,3919,3921,3945],{"className":3920,"ariaHidden":612},[611],[574,3922,3924,3927,3930,3933,3936,3939,3942],{"className":3923},[616],[574,3925],{"className":3926,"style":732},[620],[574,3928,705],{"className":3929},[625,629],[574,3931,710],{"className":3932},[739],[574,3934,598],{"className":3935},[625,629],[574,3937],{"className":3938,"style":987},[833],[574,3940,957],{"className":3941},[991],[574,3943],{"className":3944,"style":987},[833],[574,3946,3948,3951,3954],{"className":3947},[616],[574,3949],{"className":3950,"style":732},[620],[574,3952,960],{"className":3953},[625],[574,3955,719],{"className":3956},[783]," im Schritt schon benutzen."," Man muss von der Induktionsvoraussetzung ",[574,3960,3962,3981],{"className":3961},[577],[574,3963,3965],{"className":3964},[581],[583,3966,3967],{"xmlns":585},[587,3968,3969,3979],{},[590,3970,3971,3973,3975,3977],{},[596,3972,705],{},[707,3974,710],{"stretchy":709},[596,3976,598],{},[707,3978,719],{"stretchy":709},[604,3980,911],{"encoding":606},[574,3982,3984],{"className":3983,"ariaHidden":612},[611],[574,3985,3987,3990,3993,3996,3999],{"className":3986},[616],[574,3988],{"className":3989,"style":732},[620],[574,3991,705],{"className":3992},[625,629],[574,3994,710],{"className":3995},[739],[574,3997,598],{"className":3998},[625,629],[574,4000,719],{"className":4001},[783]," ausgehen und ",[574,4004,4006,4029],{"className":4005},[577],[574,4007,4009],{"className":4008},[581],[583,4010,4011],{"xmlns":585},[587,4012,4013,4027],{},[590,4014,4015,4017,4019,4021,4023,4025],{},[596,4016,705],{},[707,4018,710],{"stretchy":709},[596,4020,598],{},[707,4022,957],{},[600,4024,960],{},[707,4026,719],{"stretchy":709},[604,4028,965],{"encoding":606},[574,4030,4032,4056],{"className":4031,"ariaHidden":612},[611],[574,4033,4035,4038,4041,4044,4047,4050,4053],{"className":4034},[616],[574,4036],{"className":4037,"style":732},[620],[574,4039,705],{"className":4040},[625,629],[574,4042,710],{"className":4043},[739],[574,4045,598],{"className":4046},[625,629],[574,4048],{"className":4049,"style":987},[833],[574,4051,957],{"className":4052},[991],[574,4054],{"className":4055,"style":987},[833],[574,4057,4059,4062,4065],{"className":4058},[616],[574,4060],{"className":4061,"style":732},[620],[574,4063,960],{"className":4064},[625],[574,4066,719],{"className":4067},[783]," herleiten, nicht die Behauptung umformen, bis etwas Wahres dasteht.",[685,4070,4071,4103,4104,4195],{},[569,4072,4073,4074,4102],{},"Schritt nur für ein festes ",[574,4075,4077,4090],{"className":4076},[577],[574,4078,4080],{"className":4079},[581],[583,4081,4082],{"xmlns":585},[587,4083,4084,4088],{},[590,4085,4086],{},[596,4087,598],{},[604,4089,598],{"encoding":606},[574,4091,4093],{"className":4092,"ariaHidden":612},[611],[574,4094,4096,4099],{"className":4095},[616],[574,4097],{"className":4098,"style":1882},[620],[574,4100,598],{"className":4101},[625,629]," zeigen."," Der Schritt muss für jedes ",[574,4105,4107,4128],{"className":4106},[577],[574,4108,4110],{"className":4109},[581],[583,4111,4112],{"xmlns":585},[587,4113,4114,4126],{},[590,4115,4116,4118,4120],{},[596,4117,598],{},[707,4119,807],{},[593,4121,4122,4124],{},[596,4123,598],{},[600,4125,602],{},[604,4127,816],{"encoding":606},[574,4129,4131,4149],{"className":4130,"ariaHidden":612},[611],[574,4132,4134,4137,4140,4143,4146],{"className":4133},[616],[574,4135],{"className":4136,"style":826},[620],[574,4138,598],{"className":4139},[625,629],[574,4141],{"className":4142,"style":834},[833],[574,4144,807],{"className":4145},[838],[574,4147],{"className":4148,"style":834},[833],[574,4150,4152,4155],{"className":4151},[616],[574,4153],{"className":4154,"style":621},[620],[574,4156,4158,4161],{"className":4157},[625],[574,4159,598],{"className":4160},[625,629],[574,4162,4164],{"className":4163},[633],[574,4165,4167,4187],{"className":4166},[637,638],[574,4168,4170,4184],{"className":4169},[642],[574,4171,4173],{"className":4172,"style":647},[646],[574,4174,4175,4178],{"style":650},[574,4176],{"className":4177,"style":655},[654],[574,4179,4181],{"className":4180},[659,660,661,662],[574,4182,602],{"className":4183},[625,662],[574,4185,670],{"className":4186},[669],[574,4188,4190],{"className":4189},[642],[574,4191,4193],{"className":4192,"style":677},[646],[574,4194],{}," gelten. Bei Aussagen mit zwei Vorgängern (starke Induktion) braucht man entsprechend zwei Startwerte.",[685,4197,4198,690,4201,4315,4316,4394,4395,4529,4530,4580,4581,4686,4687,4779,4780,5071,5072,1007],{},[569,4199,4200],{},"Falscher Startwert.",[574,4202,4204,4231],{"className":4203},[577],[574,4205,4207],{"className":4206},[581],[583,4208,4209],{"xmlns":585},[587,4210,4211,4228],{},[590,4212,4213,4219,4222],{},[1643,4214,4215,4217],{},[600,4216,1366],{},[596,4218,598],{},[707,4220,4221],{},">",[1643,4223,4224,4226],{},[596,4225,598],{},[600,4227,1366],{},[604,4229,4230],{"encoding":606},"2^n > n^2",[574,4232,4234,4280],{"className":4233,"ariaHidden":612},[611],[574,4235,4237,4241,4271,4274,4277],{"className":4236},[616],[574,4238],{"className":4239,"style":4240},[620],"height:0.7035em;vertical-align:-0.0391em;",[574,4242,4244,4247],{"className":4243},[625],[574,4245,1366],{"className":4246},[625],[574,4248,4250],{"className":4249},[633],[574,4251,4253],{"className":4252},[637],[574,4254,4256],{"className":4255},[642],[574,4257,4260],{"className":4258,"style":4259},[646],"height:0.6644em;",[574,4261,4262,4265],{"style":1783},[574,4263],{"className":4264,"style":655},[654],[574,4266,4268],{"className":4267},[659,660,661,662],[574,4269,598],{"className":4270},[625,629,662],[574,4272],{"className":4273,"style":834},[833],[574,4275,4221],{"className":4276},[838],[574,4278],{"className":4279,"style":834},[833],[574,4281,4283,4286],{"className":4282},[616],[574,4284],{"className":4285,"style":1762},[620],[574,4287,4289,4292],{"className":4288},[625],[574,4290,598],{"className":4291},[625,629],[574,4293,4295],{"className":4294},[633],[574,4296,4298],{"className":4297},[637],[574,4299,4301],{"className":4300},[642],[574,4302,4304],{"className":4303,"style":1762},[646],[574,4305,4306,4309],{"style":1783},[574,4307],{"className":4308,"style":655},[654],[574,4310,4312],{"className":4311},[659,660,661,662],[574,4313,1366],{"className":4314},[625,662]," ist für ",[574,4317,4319,4345],{"className":4318},[577],[574,4320,4322],{"className":4321},[581],[583,4323,4324],{"xmlns":585},[587,4325,4326,4342],{},[590,4327,4328,4330,4332,4334,4336,4338,4340],{},[596,4329,598],{},[707,4331,1641],{},[600,4333,1366],{},[707,4335,1482],{"separator":612},[600,4337,1619],{},[707,4339,1482],{"separator":612},[600,4341,3507],{},[604,4343,4344],{"encoding":606},"n = 2, 3, 4",[574,4346,4348,4366],{"className":4347,"ariaHidden":612},[611],[574,4349,4351,4354,4357,4360,4363],{"className":4350},[616],[574,4352],{"className":4353,"style":1882},[620],[574,4355,598],{"className":4356},[625,629],[574,4358],{"className":4359,"style":834},[833],[574,4361,1641],{"className":4362},[838],[574,4364],{"className":4365,"style":834},[833],[574,4367,4369,4373,4376,4379,4382,4385,4388,4391],{"className":4368},[616],[574,4370],{"className":4371,"style":4372},[620],"height:0.8389em;vertical-align:-0.1944em;",[574,4374,1366],{"className":4375},[625],[574,4377,1482],{"className":4378},[1560],[574,4380],{"className":4381,"style":1564},[833],[574,4383,1619],{"className":4384},[625],[574,4386,1482],{"className":4387},[1560],[574,4389],{"className":4390,"style":1564},[833],[574,4392,3507],{"className":4393},[625]," falsch (z.B. ",[574,4396,4398,4429],{"className":4397},[577],[574,4399,4401],{"className":4400},[581],[583,4402,4403],{"xmlns":585},[587,4404,4405,4426],{},[590,4406,4407,4413,4415,4418,4420],{},[1643,4408,4409,4411],{},[600,4410,1366],{},[600,4412,3507],{},[707,4414,1641],{},[600,4416,4417],{},"16",[707,4419,1641],{},[1643,4421,4422,4424],{},[600,4423,3507],{},[600,4425,1366],{},[604,4427,4428],{"encoding":606},"2^4 = 16 = 4^2",[574,4430,4432,4476,4494],{"className":4431,"ariaHidden":612},[611],[574,4433,4435,4438,4467,4470,4473],{"className":4434},[616],[574,4436],{"className":4437,"style":1762},[620],[574,4439,4441,4444],{"className":4440},[625],[574,4442,1366],{"className":4443},[625],[574,4445,4447],{"className":4446},[633],[574,4448,4450],{"className":4449},[637],[574,4451,4453],{"className":4452},[642],[574,4454,4456],{"className":4455,"style":1762},[646],[574,4457,4458,4461],{"style":1783},[574,4459],{"className":4460,"style":655},[654],[574,4462,4464],{"className":4463},[659,660,661,662],[574,4465,3507],{"className":4466},[625,662],[574,4468],{"className":4469,"style":834},[833],[574,4471,1641],{"className":4472},[838],[574,4474],{"className":4475,"style":834},[833],[574,4477,4479,4482,4485,4488,4491],{"className":4478},[616],[574,4480],{"className":4481,"style":1842},[620],[574,4483,4417],{"className":4484},[625],[574,4486],{"className":4487,"style":834},[833],[574,4489,1641],{"className":4490},[838],[574,4492],{"className":4493,"style":834},[833],[574,4495,4497,4500],{"className":4496},[616],[574,4498],{"className":4499,"style":1762},[620],[574,4501,4503,4506],{"className":4502},[625],[574,4504,3507],{"className":4505},[625],[574,4507,4509],{"className":4508},[633],[574,4510,4512],{"className":4511},[637],[574,4513,4515],{"className":4514},[642],[574,4516,4518],{"className":4517,"style":1762},[646],[574,4519,4520,4523],{"style":1783},[574,4521],{"className":4522,"style":655},[654],[574,4524,4526],{"className":4525},[659,660,661,662],[574,4527,1366],{"className":4528},[625,662],") und gilt dann wieder ab ",[574,4531,4533,4550],{"className":4532},[577],[574,4534,4536],{"className":4535},[581],[583,4537,4538],{"xmlns":585},[587,4539,4540,4548],{},[590,4541,4542,4544,4546],{},[596,4543,598],{},[707,4545,1641],{},[600,4547,2485],{},[604,4549,2488],{"encoding":606},[574,4551,4553,4571],{"className":4552,"ariaHidden":612},[611],[574,4554,4556,4559,4562,4565,4568],{"className":4555},[616],[574,4557],{"className":4558,"style":1882},[620],[574,4560,598],{"className":4561},[625,629],[574,4563],{"className":4564,"style":834},[833],[574,4566,1641],{"className":4567},[838],[574,4569],{"className":4570,"style":834},[833],[574,4572,4574,4577],{"className":4573},[616],[574,4575],{"className":4576,"style":1842},[620],[574,4578,2485],{"className":4579},[625],", denn ",[574,4582,4584,4611],{"className":4583},[577],[574,4585,4587],{"className":4586},[581],[583,4588,4589],{"xmlns":585},[587,4590,4591,4608],{},[590,4592,4593,4599,4601,4604,4606],{},[1643,4594,4595,4597],{},[600,4596,1366],{},[600,4598,2485],{},[707,4600,1641],{},[600,4602,4603],{},"32",[707,4605,4221],{},[600,4607,2556],{},[604,4609,4610],{"encoding":606},"2^5 = 32 > 25",[574,4612,4614,4658,4677],{"className":4613,"ariaHidden":612},[611],[574,4615,4617,4620,4649,4652,4655],{"className":4616},[616],[574,4618],{"className":4619,"style":1762},[620],[574,4621,4623,4626],{"className":4622},[625],[574,4624,1366],{"className":4625},[625],[574,4627,4629],{"className":4628},[633],[574,4630,4632],{"className":4631},[637],[574,4633,4635],{"className":4634},[642],[574,4636,4638],{"className":4637,"style":1762},[646],[574,4639,4640,4643],{"style":1783},[574,4641],{"className":4642,"style":655},[654],[574,4644,4646],{"className":4645},[659,660,661,662],[574,4647,2485],{"className":4648},[625,662],[574,4650],{"className":4651,"style":834},[833],[574,4653,1641],{"className":4654},[838],[574,4656],{"className":4657,"style":834},[833],[574,4659,4661,4665,4668,4671,4674],{"className":4660},[616],[574,4662],{"className":4663,"style":4664},[620],"height:0.6835em;vertical-align:-0.0391em;",[574,4666,4603],{"className":4667},[625],[574,4669],{"className":4670,"style":834},[833],[574,4672,4221],{"className":4673},[838],[574,4675],{"className":4676,"style":834},[833],[574,4678,4680,4683],{"className":4679},[616],[574,4681],{"className":4682,"style":1842},[620],[574,4684,2556],{"className":4685},[625],". Der Induktionsbeweis muss deshalb mit ",[574,4688,4690,4712],{"className":4689},[577],[574,4691,4693],{"className":4692},[581],[583,4694,4695],{"xmlns":585},[587,4696,4697,4709],{},[590,4698,4699,4705,4707],{},[593,4700,4701,4703],{},[596,4702,598],{},[600,4704,602],{},[707,4706,1641],{},[600,4708,2485],{},[604,4710,4711],{"encoding":606},"n_0 = 5",[574,4713,4715,4770],{"className":4714,"ariaHidden":612},[611],[574,4716,4718,4721,4761,4764,4767],{"className":4717},[616],[574,4719],{"className":4720,"style":621},[620],[574,4722,4724,4727],{"className":4723},[625],[574,4725,598],{"className":4726},[625,629],[574,4728,4730],{"className":4729},[633],[574,4731,4733,4753],{"className":4732},[637,638],[574,4734,4736,4750],{"className":4735},[642],[574,4737,4739],{"className":4738,"style":647},[646],[574,4740,4741,4744],{"style":650},[574,4742],{"className":4743,"style":655},[654],[574,4745,4747],{"className":4746},[659,660,661,662],[574,4748,602],{"className":4749},[625,662],[574,4751,670],{"className":4752},[669],[574,4754,4756],{"className":4755},[642],[574,4757,4759],{"className":4758,"style":677},[646],[574,4760],{},[574,4762],{"className":4763,"style":834},[833],[574,4765,1641],{"className":4766},[838],[574,4768],{"className":4769,"style":834},[833],[574,4771,4773,4776],{"className":4772},[616],[574,4774],{"className":4775,"style":1842},[620],[574,4777,2485],{"className":4778},[625]," beginnen; der Schritt ",[574,4781,4783,4846],{"className":4782},[577],[574,4784,4786],{"className":4785},[581],[583,4787,4788],{"xmlns":585},[587,4789,4790,4843],{},[590,4791,4792,4804,4806,4808,4811,4817,4819,4821,4827,4829,4831,4833,4835,4837],{},[1643,4793,4794,4796],{},[600,4795,1366],{},[590,4797,4798,4800,4802],{},[596,4799,598],{},[707,4801,957],{},[600,4803,960],{},[707,4805,1641],{},[600,4807,1366],{},[707,4809,4810],{},"⋅",[1643,4812,4813,4815],{},[600,4814,1366],{},[596,4816,598],{},[707,4818,4221],{},[600,4820,1366],{},[1643,4822,4823,4825],{},[596,4824,598],{},[600,4826,1366],{},[707,4828,807],{},[707,4830,710],{"stretchy":709},[596,4832,598],{},[707,4834,957],{},[600,4836,960],{},[1643,4838,4839,4841],{},[707,4840,719],{"stretchy":709},[600,4842,1366],{},[604,4844,4845],{"encoding":606},"2^{n+1} = 2 \\cdot 2^n > 2n^2 \\geq (n+1)^2",[574,4847,4849,4902,4920,4964,5012,5033],{"className":4848,"ariaHidden":612},[611],[574,4850,4852,4855,4893,4896,4899],{"className":4851},[616],[574,4853],{"className":4854,"style":1762},[620],[574,4856,4858,4861],{"className":4857},[625],[574,4859,1366],{"className":4860},[625],[574,4862,4864],{"className":4863},[633],[574,4865,4867],{"className":4866},[637],[574,4868,4870],{"className":4869},[642],[574,4871,4873],{"className":4872,"style":1762},[646],[574,4874,4875,4878],{"style":1783},[574,4876],{"className":4877,"style":655},[654],[574,4879,4881],{"className":4880},[659,660,661,662],[574,4882,4884,4887,4890],{"className":4883},[625,662],[574,4885,598],{"className":4886},[625,629,662],[574,4888,957],{"className":4889},[991,662],[574,4891,960],{"className":4892},[625,662],[574,4894],{"className":4895,"style":834},[833],[574,4897,1641],{"className":4898},[838],[574,4900],{"className":4901,"style":834},[833],[574,4903,4905,4908,4911,4914,4917],{"className":4904},[616],[574,4906],{"className":4907,"style":1842},[620],[574,4909,1366],{"className":4910},[625],[574,4912],{"className":4913,"style":987},[833],[574,4915,4810],{"className":4916},[991],[574,4918],{"className":4919,"style":987},[833],[574,4921,4923,4926,4955,4958,4961],{"className":4922},[616],[574,4924],{"className":4925,"style":4240},[620],[574,4927,4929,4932],{"className":4928},[625],[574,4930,1366],{"className":4931},[625],[574,4933,4935],{"className":4934},[633],[574,4936,4938],{"className":4937},[637],[574,4939,4941],{"className":4940},[642],[574,4942,4944],{"className":4943,"style":4259},[646],[574,4945,4946,4949],{"style":1783},[574,4947],{"className":4948,"style":655},[654],[574,4950,4952],{"className":4951},[659,660,661,662],[574,4953,598],{"className":4954},[625,629,662],[574,4956],{"className":4957,"style":834},[833],[574,4959,4221],{"className":4960},[838],[574,4962],{"className":4963,"style":834},[833],[574,4965,4967,4971,4974,5003,5006,5009],{"className":4966},[616],[574,4968],{"className":4969,"style":4970},[620],"height:0.9501em;vertical-align:-0.136em;",[574,4972,1366],{"className":4973},[625],[574,4975,4977,4980],{"className":4976},[625],[574,4978,598],{"className":4979},[625,629],[574,4981,4983],{"className":4982},[633],[574,4984,4986],{"className":4985},[637],[574,4987,4989],{"className":4988},[642],[574,4990,4992],{"className":4991,"style":1762},[646],[574,4993,4994,4997],{"style":1783},[574,4995],{"className":4996,"style":655},[654],[574,4998,5000],{"className":4999},[659,660,661,662],[574,5001,1366],{"className":5002},[625,662],[574,5004],{"className":5005,"style":834},[833],[574,5007,807],{"className":5008},[838],[574,5010],{"className":5011,"style":834},[833],[574,5013,5015,5018,5021,5024,5027,5030],{"className":5014},[616],[574,5016],{"className":5017,"style":732},[620],[574,5019,710],{"className":5020},[739],[574,5022,598],{"className":5023},[625,629],[574,5025],{"className":5026,"style":987},[833],[574,5028,957],{"className":5029},[991],[574,5031],{"className":5032,"style":987},[833],[574,5034,5036,5039,5042],{"className":5035},[616],[574,5037],{"className":5038,"style":3139},[620],[574,5040,960],{"className":5041},[625],[574,5043,5045,5048],{"className":5044},[783],[574,5046,719],{"className":5047},[783],[574,5049,5051],{"className":5050},[633],[574,5052,5054],{"className":5053},[637],[574,5055,5057],{"className":5056},[642],[574,5058,5060],{"className":5059,"style":1762},[646],[574,5061,5062,5065],{"style":1783},[574,5063],{"className":5064,"style":655},[654],[574,5066,5068],{"className":5067},[659,660,661,662],[574,5069,1366],{"className":5070},[625,662]," funktioniert auch erst ab ",[574,5073,5075,5093],{"className":5074},[577],[574,5076,5078],{"className":5077},[581],[583,5079,5080],{"xmlns":585},[587,5081,5082,5090],{},[590,5083,5084,5086,5088],{},[596,5085,598],{},[707,5087,807],{},[600,5089,1619],{},[604,5091,5092],{"encoding":606},"n \\geq 3",[574,5094,5096,5114],{"className":5095,"ariaHidden":612},[611],[574,5097,5099,5102,5105,5108,5111],{"className":5098},[616],[574,5100],{"className":5101,"style":826},[620],[574,5103,598],{"className":5104},[625,629],[574,5106],{"className":5107,"style":834},[833],[574,5109,807],{"className":5110},[838],[574,5112],{"className":5113,"style":834},[833],[574,5115,5117,5120],{"className":5116},[616],[574,5118],{"className":5119,"style":1842},[620],[574,5121,1619],{"className":5122},[625],{"title":5124,"searchDepth":5125,"depth":5125,"links":5126},"",2,[5127,5128,5129],{"id":562,"depth":5125,"text":563},{"id":1593,"depth":5125,"text":1594},{"id":3627,"depth":5125,"text":3628},[5131,5134,5137,5139,5142,5144,5146,5149,5151],{"lhs":5132,"rhs":5133},"n^2 + (2*(n+1) - 1)","(n+1)^2",{"lhs":5135,"rhs":5136},"sum(map(1:5, f(k) = 2*k - 1))","5^2",{"lhs":5138,"rhs":2556},"1 + 3 + 5 + 7 + 9",{"lhs":5140,"rhs":5141},"(n+1)^3 - (n+1)","(n^3 - n) + 3*(n^2 + n)",{"lhs":5143,"rhs":602},"mod(4^3 - 4, 3)",{"lhs":5145,"rhs":3412},"2^3 - 2",{"lhs":5147,"rhs":5148},"2^4","4^2",{"lhs":5150,"rhs":4603},"2^5",{"lhs":5136,"rhs":2556},"n, n_0 \\in \\mathbb{Z},\\ A(n) \\text{ Aussage für jedes ganze } n \\geq n_0","\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>n\u003C\u002Fmi>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003Cmsub>\u003Cmi>n\u003C\u002Fmi>\u003Cmn>0\u003C\u002Fmn>\u003C\u002Fmsub>\u003Cmo>∈\u003C\u002Fmo>\u003Cmi mathvariant=\"double-struck\">Z\u003C\u002Fmi>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">n, n_0 \\in \\mathbb{Z},\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:0.7335em;vertical-align:-0.1944em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">n\u003C\u002Fspan>\u003Cspan class=\"mpunct\">,\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">n\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3011em;\">\u003Cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"katex-sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">0\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">∈\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:0.8833em;vertical-align:-0.1944em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathbb\">Z\u003C\u002Fspan>\u003Cspan class=\"mpunct\">,\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan> \u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>A\u003C\u002Fmi>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmi>n\u003C\u002Fmi>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">A(n)\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">A\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">n\u003C\u002Fspan>\u003Cspan class=\"mclose\">)\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan> \u003Cspan class=\"katex\">Aussage für jedes ganze\u003C\u002Fspan> \u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>n\u003C\u002Fmi>\u003Cmo>≥\u003C\u002Fmo>\u003Cmsub>\u003Cmi>n\u003C\u002Fmi>\u003Cmn>0\u003C\u002Fmn>\u003C\u002Fmsub>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">n \\geq n_0\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:0.7719em;vertical-align:-0.136em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">n\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">≥\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:0.5806em;vertical-align:-0.15em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">n\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3011em;\">\u003Cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"katex-sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">0\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",null,"md","mathematical-induction","\\bigl(A(n_0) \\wedge \\forall n \\geq n_0 : A(n) \\Rightarrow A(n+1)\\bigr) \\;\\Rightarrow\\; \\forall n \\geq n_0 : A(n)","\u003Cspan class=\"katex-display\">\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\" display=\"block\">\u003Csemantics>\u003Cmrow>\u003Cmo fence=\"true\" stretchy=\"true\" minsize=\"1.2em\" maxsize=\"1.2em\">(\u003C\u002Fmo>\u003Cmi>A\u003C\u002Fmi>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmsub>\u003Cmi>n\u003C\u002Fmi>\u003Cmn>0\u003C\u002Fmn>\u003C\u002Fmsub>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003Cmo>∧\u003C\u002Fmo>\u003Cmi mathvariant=\"normal\">∀\u003C\u002Fmi>\u003Cmi>n\u003C\u002Fmi>\u003Cmo>≥\u003C\u002Fmo>\u003Cmsub>\u003Cmi>n\u003C\u002Fmi>\u003Cmn>0\u003C\u002Fmn>\u003C\u002Fmsub>\u003Cmo>:\u003C\u002Fmo>\u003Cmi>A\u003C\u002Fmi>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmi>n\u003C\u002Fmi>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003Cmo>⇒\u003C\u002Fmo>\u003Cmi>A\u003C\u002Fmi>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmi>n\u003C\u002Fmi>\u003Cmo>+\u003C\u002Fmo>\u003Cmn>1\u003C\u002Fmn>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003Cmo fence=\"true\" stretchy=\"true\" minsize=\"1.2em\" maxsize=\"1.2em\">)\u003C\u002Fmo>\u003Cmtext>  \u003C\u002Fmtext>\u003Cmo>⇒\u003C\u002Fmo>\u003Cmtext>  \u003C\u002Fmtext>\u003Cmi mathvariant=\"normal\">∀\u003C\u002Fmi>\u003Cmi>n\u003C\u002Fmi>\u003Cmo>≥\u003C\u002Fmo>\u003Cmsub>\u003Cmi>n\u003C\u002Fmi>\u003Cmn>0\u003C\u002Fmn>\u003C\u002Fmsub>\u003Cmo>:\u003C\u002Fmo>\u003Cmi>A\u003C\u002Fmi>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmi>n\u003C\u002Fmi>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">\\bigl(A(n_0) \\wedge \\forall n \\geq n_0 : A(n) \\Rightarrow A(n+1)\\bigr) \\;\\Rightarrow\\; \\forall n \\geq n_0 : A(n)\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:1.2em;vertical-align:-0.35em;\">\u003C\u002Fspan>\u003Cspan class=\"mopen\">\u003Cspan class=\"delimsizing size1\">(\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">A\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">n\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3011em;\">\u003Cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"katex-sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">0\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mclose\">)\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003Cspan class=\"mbin\">∧\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:0.8304em;vertical-align:-0.136em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">∀\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">n\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">≥\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:0.5806em;vertical-align:-0.15em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">n\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3011em;\">\u003Cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"katex-sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">0\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">:\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">A\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">n\u003C\u002Fspan>\u003Cspan class=\"mclose\">)\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">⇒\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">A\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">n\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003Cspan class=\"mbin\">+\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:1.2em;vertical-align:-0.35em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">1\u003C\u002Fspan>\u003Cspan class=\"mclose\">)\u003C\u002Fspan>\u003Cspan class=\"mclose\">\u003Cspan class=\"delimsizing size1\">)\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">⇒\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:0.8304em;vertical-align:-0.136em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">∀\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">n\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">≥\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:0.5806em;vertical-align:-0.15em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">n\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3011em;\">\u003Cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"katex-sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">0\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">:\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">A\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">n\u003C\u002Fspan>\u003Cspan class=\"mclose\">)\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>","university",{},true,"\u002Fde\u002Fformulas\u002Fmath\u002Fdiscrete-math\u002Fvollstaendige-induktion",[],[5165,5166,5167],"gauss-sum","sum-of-squares","fibonacci-binet",{"title":548,"description":5124},"de\u002Fformulas\u002Fmath\u002Fdiscrete-math\u002Fvollstaendige-induktion","Gilt eine Aussage für einen Startwert und folgt sie stets von n auf n+1, dann gilt sie für alle n ab dem Startwert.",[5172,5173,5174,5175],"beweisverfahren","induktionsanfang","induktionsschritt","natürliche zahlen",[5177,5180,5183,5186],{"symbol":911,"meaning":5178,"symbolHtml":5179},"Aussage, die von der ganzen Zahl n abhängt","\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>A\u003C\u002Fmi>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmi>n\u003C\u002Fmi>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">A(n)\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">A\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">n\u003C\u002Fspan>\u003Cspan class=\"mclose\">)\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",{"symbol":607,"meaning":5181,"symbolHtml":5182},"Startwert, ab dem die Aussage gelten soll (meist 0 oder 1)","\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmsub>\u003Cmi>n\u003C\u002Fmi>\u003Cmn>0\u003C\u002Fmn>\u003C\u002Fmsub>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">n_0\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:0.5806em;vertical-align:-0.15em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">n\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3011em;\">\u003Cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"katex-sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">0\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",{"symbol":722,"meaning":5184,"symbolHtml":5185},"Induktionsanfang, die Aussage für den Startwert","\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>A\u003C\u002Fmi>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmsub>\u003Cmi>n\u003C\u002Fmi>\u003Cmn>0\u003C\u002Fmn>\u003C\u002Fmsub>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">A(n_0)\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">A\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">n\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3011em;\">\u003Cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"katex-sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">0\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mclose\">)\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",{"symbol":5187,"meaning":5188,"symbolHtml":5189},"A(n) \\Rightarrow A(n+1)","Induktionsschritt, aus der Gültigkeit für n folgt die für n+1","\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>A\u003C\u002Fmi>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmi>n\u003C\u002Fmi>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003Cmo>⇒\u003C\u002Fmo>\u003Cmi>A\u003C\u002Fmi>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmi>n\u003C\u002Fmi>\u003Cmo>+\u003C\u002Fmo>\u003Cmn>1\u003C\u002Fmn>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">A(n) \\Rightarrow A(n+1)\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">A\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">n\u003C\u002Fspan>\u003Cspan class=\"mclose\">)\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">⇒\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">A\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">n\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003Cspan class=\"mbin\">+\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">1\u003C\u002Fspan>\u003Cspan class=\"mclose\">)\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>","NYXyTlJ1K-_OjlwrPJXvy-ylFjPsdqxMwifI3zgjHJ8",[5192,5195,5198],{"stem":5193,"title":5194,"category":187},"de\u002Fformulas\u002Fmath\u002Fdiscrete-math\u002Fformel-von-binet","Formel von Binet",{"stem":5196,"title":5197,"category":412},"de\u002Fformulas\u002Fmath\u002Fsequences-series\u002Fgausssche-summenformel","Gaußsche Summenformel",{"stem":5199,"title":5200,"category":412},"de\u002Fformulas\u002Fmath\u002Fsequences-series\u002Fsumme-der-quadratzahlen","Summe der Quadratzahlen",1793824276913]