Question

已知{an}满足an=(3n+1)2^（n+1）,求Sn,用错位相减法

Answer 1

依题意得

Sn=4×2^2+7×2^3+10×2^4+…+(3n+1)×2^(n+1)

所以等式左右乘以等比数列的公比2,得：

2Sn=4×2^3+7×2^4+10×2^5+…+(3n+1)×2^(n+2)

所以2Sn-Sn=Sn=-4×2^2-[3×2^3+3×2^4+…+3×2^(n+1)]+(3n+1)×2^(n+2)

=-16-3[2^3+2^4+2^5+…+2^(n+1)]+(3n+1)×2^(n+2)

=-16-3[-2^3+2^(n+2)]+(3n+1)×2^(n+2)

整理得

=(3n-2)×2^(n+2)+8

已验算