递推公式An+1=2An+2ⁿ（PQ为常数）,求｛An｝

Question

递推公式An+1=2An+2ⁿ（P Q为常数）,求｛An｝

Answer 1

An+1=2An+2ⁿ
两边同时除以2^(n+1)
则A(n+1)/2^(n+1)=A(n)/2^n+1/2
即 A(n+1)/2^(n+1)-A(n)/2^n=1/2
∴ {A(n)/2^n}是等差数列.
公差是1/2,首项是A1/2 (你题目中缺少A1)
∴ A(n)/2^n=A1/2+(1/2)*(n-1)=[A1+(n-1)]/2
∴ A(n)=[A1+(n-1)]*2^(n-1)