A(n+1)=3An+n的三次方,怎么构造等比数列?

用待定系数法啊.
A(n+1) = 3An + n^3,构造等比数列会具有下面的形式
A(n+1) + h(n+1)^3 + i(n+1)^2 + j(n+1) + k = 3(An + hn^3 + in^2 + jn + k),h,i,j,k是待定的系数
整理上式得到
A(n+1) + hn^3 + (3h + i)n^2 + (3h + 2i + j)n + (h + i + j + k )
= 3An + 3hn^3 + 3in^2 + 3jn + 3k,继续化简
A(n+1) = 3An + 2hn^3 + (2i - 3h)n^2 + (2j - 2i - 3h)n + (2k - h - i -j)
比较系数可得
2h = 1
2i - 3h = 0
2j - 2i - 3h = 0
2k - h - i - j = 0
依次解出
h = 1/2
i = 3/4
j = 3/2
k = 11/8
所以数列{An+1/2 n^3 + 3/4 n^2 + 3/2 n + 11/8}是等比数列

A(n+1)=3An+n^3
两边同加x得
A(n+1)+x=3An+n^3+x=3[An+(n^3+x)/3]
令x=(n^3+x)/3
解得x=n^3/2
A(n+1)+n^3/2=3[An+n^3/2]
[A(n+1)+n^3/2]/[An+n^3/2]=3
数列{An+n^3/2}是公比为3的等比数列