(1)1×2+2×3+3×4+…+n(n+1)=3分之1×____ (2)1×2×3+2×3×4+3×4×5+…+7×8×9=4分之1×____=____.

题目

(1)1×2+2×3+3×4+…+n(n+1)=3分之1×____ (2)1×2×3+2×3×4+3×4×5+…+7×8×9=4分之1×____=____.

答案

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
1×2×3+2×3×4+3×4×5……+7×8×9
=1/4×7×8×9×10
=1260