题目

若a1=3,an+1=3an+3^n+1 1)设bn=an/3^n 证明:数列{bn}是等比数列 (2)求数列{an}的前n项和Sn

提问时间：2020-11-03

答案

(1)
a(n+1)=3an+3^(n+1)
a(n+1)/3^(n+1)-an/3^n= 1
{an/3^n} 是等差数列,d=1
an/3^n-a1/3^1=n-1
an/3^n= n
an = n.3^n
(2)
bn = an/3^n = n
{bn}是等差数列
let
S = 1.3^1+2.3^2+...+n.3^n (1)
3S = 1.3^2+2.3^3+...+n.3^(n+1) (2)
(2)-(1)
2S = n.3^(n+1)-(3+3^2+...+3^n)
=n.3^(n+1)- (3/2)(3^n-1)
S = 3 + (6n-3).3^n
Sn = a1+a2+...+an = S = 3 + (6n-3).3^n