已知数列an满足a1+2a2+3a3+……+nan=n(n+1)(n+2),则它的前n项和Sn=?

题目

已知数列an满足a1+2a2+3a3+……+nan=n(n+1)(n+2),则它的前n项和Sn=?
（9n+3n²）/2）

答案

设a1+2a2+3a3+……+nan=Tn
则Tn=n(n+1)(n+2)
T(n-1)=n(n-1)(n+1)
nan=Tn-T(n-1)=3n(n+1)
an=3(n+1)
Sn=3(1+1)+3(2+1)+...+3(n+1)
=3n(n+3)/2=(3n^2+9n)/2