1x2x3+2x3x4+3x4x5+4x5x6+...+n(n+1)(n+2)=

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1x2x3+2x3x4+3x4x5+4x5x6+...+n(n+1)(n+2)=1x2x3+2x3x4+3x4x5+4x5x6+...+n(n+1)(n+2)=1x2x3+2x3x4+3x4x5+4x

1x2x3+2x3x4+3x4x5+4x5x6+...+n(n+1)(n+2)=
1x2x3+2x3x4+3x4x5+4x5x6+...+n(n+1)(n+2)=

1x2x3+2x3x4+3x4x5+4x5x6+...+n(n+1)(n+2)=
原式=1/4（-0*1*2*3+1*2*3*4）+1/4（-1*2*3*4+2*3*4*5）+……+1/4[-（n-1)n(n+1)(n+2)+n(n+1)(n+2)(n+3)]
=1/4[-0*1*2*3*4+n(n+1)(n+2)(n+3)]
=n(n+1)(n+2)(n+3)/4