# Prove this sum of binomial terms using induction.

Here's the problem stumping me today:

Let $n \in \mathbb{N}$ and $r \in \mathbb{N}$ such that $r \leq n$, and prove using induction that $\binom{n+1}{r+1} = \sum\limits_{i=r}^n \binom{i}{r}$.

I've setup the basics of my inductive proof, but I'm struggling with the induction step.

Could anyone point me in the right direction?

Hint: By Pascal's Identity, we have: $$\binom{n}{r+1} + \binom{n}{r} = \binom{n+1}{r+1}$$ Note that this is equivalent to: $$\binom{k+1}{r+1} + \binom{k+1}{r} = \binom{k+2}{r+1}$$