MT【312】特征根法求数列通项

(2016清华自招领军计划37题改编）

设数列$\{a_n\}$满足$a_1=5,a_2=13,a_{n+2}=\dfrac{a^2_{n+1}+6^n}{a_n}$则下面不正确的是(      )
A.$a_{n+2}=5a_{n+1}-6a_n$

B.$\{a_n\}$中的项都是整数

C.$a_n>4^n$

D.$\{a_n\}$中与2015最接近的项为$a_7$

答案:C

提示:$a_{n+3}a_{n+1}-a_{n+2}^2=6^{n+1}=6(a_{n+2}a_n-a_{n+1}^2)$
易得$\dfrac{a_{n+3}+6a_{n+1}}{a_{n+2}}=\dfrac{a_{n+2}+6a_{n}}{a_{n+1}}=\dfrac{a_3+6a_1}{a_2}=5$
故$a_{n+2}=5a_{n+1}-6a_n$特征根法易知$a_n=2^n+3^n$故$a_6=793,a_7=2315$