问一道高一数列的题目,a1=1,a(n+1)=2an+2^n,求an,
问题描述：
问一道高一数列的题目,a1=1,a(n+1)=2an+2^n,求an,
答
a1=1
a2=2x1+2^1=4=2x2^1
a3=2x4+2^2=12=3x2^2
a4=2x12+2^3=32=4x2^3
……
an=nx2^(n-1)
答
将两边同时除以2^n+1,得到一个新数列，为等差数列。最后an=2^n*n/2
答
a(n+1)=2an+2^n两边同时除以2^n+1a(n+1)/(2^n+1)=an/(2^n)-1a(n+1)/(2^n+1)-an/(2^n)=1所以｛an/(2^n)｝是以1/2为首项,1为公差的等差数列所以an/(2^n)=1/2+(n-1)*(1/2)=n/2所以an=(n/2)*(2^n)=n*2^(n-1)...