Mathematical Induction and telescoping series Use a suitable telescoping sum to find a simpler expression for the sum $1^4+2^4+...+n^4$ where $n\in \mathbb{N}$
Prove by mathematical induction

The suitable telescoping sum might be
$$\sum_{k=1}^n \left[(k+1)^5-k^5\right]=(n+1)^5-1.$$
By expanding the left hand side and recalling formulas for $\sum k$, $\sum k^2$ and $\sum k^3$, you can solve for the desired sum.
