设数列an满足a（n+1）=3an+2的n次方 且a1,a2+5,a3成等差数列

题目

设数列an满足a（n+1）=3an+2的n次方 且a1,a2+5,a3成等差数列
（1）求a1的值 （2）求证数列｛a（n+2）｝是等比数列 并且求数列an的通项公式.

答案

a(n+1) = 3an +2
a(n+1) +1 = 3(an + 1)
{an +1}是等比数列,q=3
an + 1 = 3^(n-1) .(a1+1)
an = -1+ (a1+1).3^(n-1)
a2 =-1+ 3(a1+1) = 3a1+2
a3 =-1+ 9(a1+1) = 9a1+8
a1,a2+5,a3成等差数列
a1+a3 = 2(a2+5)
a1+9a1+8 = 2(3a1+7)
4a1=6
a1= 3/2
an =-1+ (a1+1).3^(n-1)
= -1+ (5/2).3^(n-1)