I am trying to use induction to prove, for every positive integer $k$, that
$$\sum_{j=1}^{n} {j^{k}} = \frac{n^{k+1}}{k+1} +\frac{n^k}{2} + P_{k-1}(n)$$
where $P_{k-1}$ is a polynomial of degree at most $k-1$.
In order to do the inductive step, I assume that the statement is true for $k$ and use that to prove the statement for $k+1$.
In order to do this I would need to express $\sum_{j=1}^{n} {j^{k+1}}$ in terms of $\sum_{j=1}^{n} {j^{k}}$. How would I do this?