若数列an满足a1=1,an+a(n+1)=(1/4)^n ,设Sn=a1+4a2+4^2a3+……+4^（n-1）

来源：学生作业帮编辑：作业帮分类：数学作业时间：2024/07/19 00:43:09

若数列an满足a1=1,an+a(n+1)=(1/4)^n ,设Sn=a1+4*a2+4^2*a3+……+4^（n-1）*an,
类比课本中推导等比数列前几项和公式的方法,可求得5*Sn-4^n*an=___.

Sn=a1+4a2+4^2a3...+4^(n-1)an
4Sn=4a1+4^2a2+4^3a3...+4^nan
5Sn=[a1+4a2+4^2a3...+4^(n-1)an]+[4a1+4^2a2+4^3a3...+4^nan]
=a1+{4(a1+a2)+4^2(a2+a3)+...+4^(n-1)*[a(n-1)+an]}+4^nan
=(1+4^nan)+{4*(1/4)^1+(4^2)*(1/4)^2+...+[4^(n-1)]*[(1/4)^(n-1)]}
=1+4^nan+(n-1)*1=4^nan+n
于是5Sn-4^nan=n

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