Induction on $S(n,k)$ Consider $$S(n,k):= 1^{k}+\ldots+n^{k}.$$
I have to find a formula for $$S(n,3).$$
After trying for some n values I found out that
$$S(1,3)=1^{3}=1^{2}$$
$$S(2,3)=1^{2} +2^{2}=3^{2}$$
$$S(3,3)=1^{2} +2^{2}  +3^{3}=6^{2}$$
$$S(4,3)=1^{2} +2^{2} +3^{3}+4^{3}=10^{2}$$
$$S(5,3)=1^{2} +2^{2} +3^{3}+4^{3}+5^{3}=15^{2}$$
Which led to the conjecture that $$S(n,3)=\left(1+\ldots+n\right)^{2}$$
I know for a fact that $$S(n,1)=\frac{n^{2}}{2}+\frac{n}{2}$$
Therefore,
$$S(n,3)=\left(\frac{n^{2}}{2}+\frac{n}{2}\right)^2$$
But when trying to prove the formula using induction I get stuck on the inductive step.
$$S(k+1,3)=1^{3}+\ldots+k^{3}+(k+1)^3=\left(\frac{(k+1)^{2}}{2}+\frac{k+1}{2}\right)^2$$
I don't know how to proceed. I have tried manipulating the RHS, but it's been fruitless.

On inductive step we can use
$$1^3+2^3+\cdots +k^3+(k+1)^3 = (1+2+\cdots +k)^2 +(k+1)^3 =(1+2+\cdots +k+(k+1))^2$$
Last equality is same with
$$\left( \frac{k(k+1)}{2} \right)^2 +(k+1)^3 = \left( \frac{(2+k)(k+1)}{2} \right)^2$$
and can be easy checked.
