Question

1×2+2×3+3×4+……n×(n+1)=( ) 填公式

Answer 1

证明过程如下

∵n(n+1)=n^2+n

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

=(1+2+3+……+n)+(1^2+2^2+3^2+……+n^2)

=n(n+1)/2+(1^2+2^2+3^2+……+n^2)

S（n)=n(n+1)(2n+1)/6

s=1^2+2^2+...+n^2

=n(n+1)(2n+1)/6

=(n^2+n)(2n+1)/6

=(2n^3+3n^2+n)/6

∴1×2+2×3+3×4+……n×(n+1)=(2n^3+3n^2+n)/6