求等差数列的立方和公式

Question

求等差数列的立方和公式

Answer 1

a(n)=a(1)+(n-1)d
[a(n)]^3=[a(1)+(n-1)d]^3=[a(1)]^3+3(n-1)d[a(1)]^2+3(n-1)^2d^2 a(1)+(n-1)^3d^3
[a[1]]^3+a[2]^3+.+a[n]^3
=n[a(1)]^3
+3d[a(1)]^2 [0+1+2+.(n-1)]
+3d^2 a(1) [0^2+1^2+.+(n-1)^2]
+d^3 [0^3+1^3+.+(n-1)^3]
=n[a(1)]^3+3d[a[1]]^2 n(n-1)/2+ n[a(1)]^3*(n-1)n(2n-1)/2+d^3*n^2(n-1)^2/4