Question

利用积分定义和公式1^3+2^3+。。。+n…3=[n（n+1）／2]^2 求详细过程，谢谢！

Answer 1

1+2+3+..+n = n(n+1)/2

n^2 = n(n+1) -n
= (1/3) [ n(n+1)(n+2) - (n-1)n(n+1) ] -n
1^2+2^2+3^2+..+n^2 = (1/3) n(n+1)(n+2) - n(n+1)/2
= (1/6)n(n+1)(2n+1)

n^3 = n(n+1)(n+2) -3n^2 -2n
= (1/4)[n(n+1)(n+2)(n+3) -(n-1)n(n+1)(n+2)] -3n^2 -2n
1^3+2^3+3^3+..+n^3
= (1/4)n(n+1)(n+2)(n+3) - 3(1/6)n(n+1)(2n+1) -2[n(n+1)/2]
= (1/4)n(n+1)[ (n+2)(n+3) - 2(2n+1) -4 ]
=(1/4)n(n+1)( n^2+n)
= [n(n+1)/2]^2