1×2+2×3+3×4+……n×(n+1)=( ) 填公式

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1×2+2×3+3×4+……n×(n+1)=( ) 填公式
1×2+2×3+3×4+……n×(n+1)=( ) 填公式

1×2+2×3+3×4+……n×(n+1)=( ) 填公式
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

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