Question

1cube+2cube+3cube+4cube=?square

Answer 1

for nth term to find ..Proof by mathematical induction.

When n = 1 the result is clear, 13 = 12

Assume the result is true for n = k, that is

13 + 23 + 33 + 43 + ... k3 = (1 + 2 + 3 +...k)2Let n = k + 1, then   13 + 23 + 33 + 43 + ... n3   = 13 + 23 + 33 + 43 + ... + (k+1)3   = (1 + 2 + 3 + ... + k)2 + (k+1)3   = (k(k+1)/2)2 + (k+1)3   = (k2 (k+1)2)/4 + (k+1)3   = (k+1)2/4 (k2 + 4k + 4)   = ((k+1)2 (k+2)2)/4   = ((k+1)(k+2)/2)2   = (1 + 2 + 3 + ... + (k+1))2

Thus 13 + 23 + 33 + 43 + ... n3 = (1+2+3+...n)2for all positive intergers n.