Question

how to prove it for all positive integers n​

Answer 1

Answer:

Suppose you have proven that

13+23+…+n3=(1+2+…+n)2

Note that using the binomial theorem

(1+2+⋯+n+(n+1))2=(1+2+⋯+n)2+2(n+1)(1+2+⋯+n)+(n+1)2

One can use at this point that 1+⋯+n=n(n+1)2 so that the above becomes

(1+2+⋯+n+(n+1))2=(1+2+⋯+n)2+n(n+1)2+(n+1)2=(1+2+⋯+n)2+(n+1)(n+1)2=(1+2+⋯+n)2+(n+1)3

and induction kicks in.