help!1*2*3+2*3*4+3*4*5+···+100*101*102

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help!1*2*3+2*3*4+3*4*5+···+100*101*102
help!
1*2*3+2*3*4+3*4*5+···+100*101*102

help!1*2*3+2*3*4+3*4*5+···+100*101*102
n(n+1)(n+2)=n^3+3n^2+2n
1×2×3+2×3×4+3×4×5+……n(n+1)(n+2)
=(1^3+2^3+……+n^3)+3*(1^2+2^2+……+n^2)+2*(1+2+……+n)
1^3+2^3+……+n^3=[n(n+1)/2]^2
1^2+2^2+……+n^2=n(n+1)(2n+1)/6
1+2+……+n=n(n+1)/2
所以原式=[n(n+1)/2]^2+3n(n+1)(2n+1)/6+2n(n+1)/2
=n^2(n+1)^2/4+n(n+1)(2n+1)/2+n(n+1)
=[n(n+1)/4]*[n(n+1)+2(2n+1)+4]
=[n(n+1)/4]*(n^2+5n+6)
=n(n+1)(n+2)(n+3)/4
然后代入就可以了

1900

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