Question

Using mathematical induction show that 1^3+2^3+3^3...+n^3=1/4n^2(n+1)^2 for all n € Ñ

Answer 1

test for n=1. 1^3 = 1^2*2^2/4 = 1 ✅

assume true for n=k, test for n=k+1
1^3 + ... + k^3 + (k+1)^3 = k^2(k+1)^2/4 + (k+1)^3
= (k^4 + 2k^3 + k^2 + 4k^3 + 12k^2 + 12k + 4)/4
= (k^4 + 6k^3 + 13k^2 + 12k + 4)/4
= (k+1)^2 * (k+2)^2 / 4
QED