Question

数列{an}中，an=(2n+1)×3^n,求Sn?

Answer 1

a1=3*3
Sn=3*3+5*3^2+7*3^3+...+(2n+1)×3^n
3Sn=3*3^2+5*3^3+7*3^4+...+(2n+1)*3^(n+1)
3Sn-Sn=(2n+1)*3^(n+1)-3*3-2*(3^2+3^3+3^4+...+3^n)
=(2n+1)*3^(n+1)-9-2*3*[3+3^2+3^3+3^4+...+3^(n-1)]
=(2n+1)*3^(n+1)-9-6*3*(1-3^n)/(1-3)
=(2n+1)*3^(n+1)-3^(n+2)
=(2n-2)*3^(n+1)
Sn=(n-1)*3^(n+1)

Answer 2

错位相减法
Sn=3X3+5X3^2+…+(2n+1)×3^n
3Sn= 3X3^2+5X3^3+…+(2n-1)×3^n+(2n+1)×3^(n+1)
-2Sn=9+2X3^2+…+2X3^n-(2n+1)×3^(n+1)
=9+3^(n+1)-9-(2n+1)×3^(n+1)
=-2n×3^(n+1)