Question

简算请你找出1*1*1+2*2*2+3*3*3+.....+N*N*N的公试

算出1*1*1+2*2*2+3*3*3+.....+100*100*100的值

Answer 1

1³+2³+3³+....+(n-1)³+n³=[n^2(n+1)^2]/4

1*1*1+2*2*2+3*3*3+.....+100*100*100
=(100*101)²/4
=25502500

Answer 2

1*1*1+2*2*2+3*3*3+.....+N*N*N=(1+2+....+n)^2=n^2*(n+1)^2/4
1*1*1+2*2*2+3*3*3+.....+100*100*100
=100^2*101^2/4=25502500

Answer 3

( n + 1 )^4 = n^4 + 4 * n^3 + 6 * n^2 + 4 * n + 1
n^4 = ( n - 1 )^4 + 4 * ( n - 1 )^3 + 6 * ( n - 1 )^2 + 4 * ( n - 1 ) + 1
.
.
.
2^4 = 1^4 + 4 * 1^3 + 6 * 1^2 + 4 * 1 + 1

等式两边分别相加
( n + 1 )^4 = 1^4 + 4 * ∑( k = 1 -> n ) k^3 + 6 * ∑( k = 1 -> n ) k^2 + 4 * ∑( k = 1 -> n ) k + n

∑( k = 1 -> n ) k^3
= ( n + 1 )^4 - 1^4 - 6 * ∑( k = 1 -> n ) k^2 - 4 * ∑( k = 1 -> n ) k - n
= ( n + 1 )^4 - 1^4 - 6 * n / 6 * ( n + 1 ) * ( 2n + 1 ) - 4 * n / 2 * ( n + 1 ) - n
= n^2 * ( n + 1 )^2 / 4

当 n = 100时,
n^2 * ( n + 1 )^2 / 4 = 25502500

Answer 4

11³+2³+3³+……+(n-1)³+n³=(1/4)(n-1)²(n)²；

1³+2³+3³+……+99³+100³=(1/4)*100²*101²=25502500

Answer 5

={(1+N)N/2}的平方
=25502500