如何证明1x2+2x3+…+n(n+1)=n(n+1)(n+2)/3顺便再证明一下1x2+2x3+…+n(n+1)（n+2）=n(n+1)(n+2)(n+3)/4

Question

如何证明1x2+2x3+…+n(n+1)=n(n+1)(n+2)/3
顺便再证明一下1x2+2x3+…+n(n+1)（n+2）=n(n+1)(n+2)(n+3)/4

Answer 1

证明1x2+2x3+…+n(n+1)=n(n+1)(n+2)/3
1x2+2x3+…+n(n+1)=1x(1+1)+2x(2+1)+.+n(n+1)
=(1^2+2^2+.+n^2)+(1+2+.+n)
=n(n+1)(2n+1)/6 + n(n+1)/2
=n(n+1)(n+2)/3
证明1x2+2x3+…+n(n+1)（n+2）=n(n+1)(n+2)(n+3)/4是错的,我想应该是证明1x2x3+2x3x4+.+n(n+1)(n+2)=n(n+1)(n+2)(n+3)/4
若是证明1x2x3+2x3x4+.+n(n+1)(n+2)=n(n+1)(n+2)(n+3)/4
因为n(n+1)(n+2)=n^3+3n^2+2n
所以1x2x3+2x3x4+.+n(n+1)(n+2)
=(1^3+2^3+...+n^3) + 3(1^2+2^2+...+n^2) + 2(1+2+...+n)
=n^2(n+1)^2/4 + n(n+1)(2n+1)/2 + n(n+1)
=n(n+1)(n+2)(n+3)/4