特征方程法求数列的通项公式
九江市教研室 林健航 求数列通项公式的方法很多,利用特征方程的特征根的方法是求一类数列通
项公式的一种有效途径.
aab,,*na,cadbcnN,,,0,,1.已知数列满足......? 其中. a,,n,1ncad,,n
axb,定义1:方程为?的特征方程,该方程的根称为数列的特征根,记为. x,,,,a,,ncxd,
aa,,,,ac,,nn,1定理1:若,,,,,,a且,则. ,,,1aaca,,,,,,nn,1
,,axbadb2证明: xcxdaxb,,,,,,,,,,,()0,,,,,cxdcc,
?,,,,,dacbc(),,,,,
aab,n,,acadaabcadacabd,,,,,,,,()()()(),,,,nnnnn,1 ?,,, aab,naaabcadacabd,,,,,,,()()()()nnnn,,,,,1,,cad,n
()[()]()()acacaccacaac,,,,,,,,,,,,,,,,,,nn ,, ()[()]()()acacaccacaac,,,,,,,,,,,,,,,,,,nn
a,,ac,,n ,, 证毕 aca,,,,n
121c定理2: 若,,,,a,,且,则. ad,,01aada,,,,,nn,1
2证明: dacbc,,,,2,,,
cadcad,,11nn ?,,, aab,n,,,,aaabcadacabd,,,,,,,()()()nnnn,1,,ncad,
caaccaaccaac,,,,,,222,,,nnn ,,, 22ad,()(2)()()acacacaca,,,,,,,,,,,,nn()a,,n2
2242(2)2()()caaccaacdcaad,,,,,,,,,,,nnn,,, ()()()()()()adaadaada,,,,,,,,,nnn
21c,, 证毕 ,,,adan
例1: (09?江西?理?22)各项均为正数的数列aaab,,,,,且对满足amnpq,,,,,12n
aa,aa,pqmn的正数都有. ,mnpq,,,,,,,aaaa(1)(1)(1)(1)mnpq
14(1)当a时,求通项;(2)略. ab,,,n25
aa,aaaa,,aa,pq121nn,mn解:由,得 ,(1)(1)(1)(1),,,,aaaa,,,,aaaa(1)(1)(1)(1)121nn,mnpq
21a,14n,1将a,代入上式化简得 ab,,,na,225n,1
21x,考虑特征方程得特征根 x,x,,1x,2
21a,n,1,1aaa,,,1211nnn,,11所以,,, 21a,n,1aa,,131nn,1,1a,2n,1
,,a,1a,111n1所以数列,,是以为首项,公比为的等比数列 ,,a,133a,11n,,
na,111131,nn,1n故,,,,,()()a, 即 nna,133331,n
1*例2:已知数列2,2,aanN,,,,a满足,求通项. a,,1nnna1n,
1解: 考虑特征方程得特征根 x,,2x,1x
a1111n,1,,,,,1 11,,,aaa111nnn,,11,,,(2)11nn,,11aa
,,11所以数列,1是以为首项,公差为1的等差数列 ,,a,11a,1n,,
1n,1故,n 即a, n1a,nn
acaca,,cc,已知数列满足......? 其中为常数,且a,,nnn,,211212n
*cnN,,0,. 22.
2定义2:方程xcxc,,,,,为?的特征方程,该方程的根称为数列的特征根,记为. a,,1212n
abb,,,,,11122nn定理3:若abb,,,,,,,bb,,则,其中常数,且满足. ,n1122121222abb,,,,,21122
abb,,(),,112n定理4: 若abbn,,(),,,,,,bb,,则,其中常数,且满足. n,1212122abb,,(2),,212例3:已知数列aaaaa,,,,1,2cos,2cos,,a满足,求通项. a,,1221nnn,,nn
2解: 考虑特征方程xx,,2cos1,,,,,,,,,,,cossin,cossinii得特征根 12
nn 则abibibbnibbn,,,,,,,,(cossin)(cossin)()cos()sin,,,,,, n121212
bb,,0,121()cos()sin,,,,bbibb,,,sinn,,1212 其中 ,,,a1,,nibb(),,2cos()cos2()sin2,,,,,,,bbibbsin,12,1212,sin,,例4:已知数列aaaaa,,,,2,8,44a满足,求通项. a,,1221nnn,,nn
2解: 考虑特征方程xx,,44得特征根 ,,2
n 则abbn,,()2 n12
2()20bbb,,,,,121n 其中,,,an2 ,,n4(2)81bbb,,,,,122
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