已知数列{An}满足a1=1,且An=2An-1+2^n(n大于等于2且n属于整数）.求数列的通项公式?

题目

已知数列{An}满足a1=1,且An=2An-1+2^n(n大于等于2且n属于整数）.求数列的通项公式?

答案

an=2(an-1+2^n)-2^n
→(an+2^n)/(an-1+2^n)=2
→{an+2^n}是首项为3,q=2的等比数列
→an+2^n=3*2^(n-1)
→an=3*2^(n-1)-2^n