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

问题描述:

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

∵a1+2a2+3a3+…+nan=n(n+1)(n+2),①
∴a1+2a2+3a3+…+(n-1)an-1=(n-1)n(n+1),②
①-②,得nan=3n(n+1),
∴an=3n+3.
∴Sn=a1+a2+a3+…+an
=(3×1+3)+(3×2+3)+(3×3+3)+…+(3n+3)
=3(1+2+3+…+n)+3n
=

n(n+1)
2
+3n
=
3n2+9n
2

故答案为:
3n2+9n
2