数列｛an｝满足a1=1,a2=3,an+2-4an+1+4an=0

题目

数列｛an｝满足a1=1,a2=3,an+2-4an+1+4an=0
数列满足A1=1,A2=3.并且An+2-4An+1+4An=0.
1、证明{An+1-2An}是等比数列

答案

设队列{An+1-2An}={Bn}
∵An+2-4An+1+4An=0
∴A(n+2)-2A(n+1)=2A(n+1)-4An
[A(n+2)-2A(n+1)]/[A(n+1)-2An]=2
∴B(n+1)/Bn
=[A(n+2)-2A(n+1)]/[A(n+1)-2An]=2
∴{An+1-2An}是等比数列