在数列﹛an﹜中,an=2an-1+2的n次方,a1=5,求an的通项公式

Question

在数列﹛an﹜中,an=2an-1+2的n次方,a1=5,求an的通项公式

Answer 1

An = 2A(n - 1) + 2^n
An/2^n = 2A(n - 1)/2^n + 1
An/2^n - A(n - 1)/2^(n - 1) = 1
即An/2^n是以1为公差的等差数列
根据定义Tn = 首项 + (n - 1)(公差)、其中Tn为An/2^n,首项为(A1)/2^1 = (A1)/2
所以An/2^n = (A1)/2^1 + (n - 1)(1) = (A1)/2 + n - 1 = 5/2 - 1 + n = 3/2 + n
得An = (3/2 + n) * 2^n