求s的值; s=1*2*3+2*3*4+3*4*5+……+n*(n+1)*(n+2)

2024-11-06 00:48:34
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回答1:

如果是简单数列问题可以这样做 n(n+1)(n+2)=n^3+3n^2+2n 1^3+2^3+……+n^3=[n(n+1)/2]^2=n^2(n+1)^2/4 1^2+2^2+……+n^2=n(n+1)(2n+1)/6 1+2+……+n=n(n+1)/2 所以1*2*3+2*3*4+……+n(n+1)(n+2) =n^2(n+1)^2/4+3*n(n+1)(2n+1)/6+2*n(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+n+4n+2+4) =[n(n+1)/4](n^2+5n+6) =n(n+1)(n+2)(n+3)/4