Given $s,n,k$ such that $n\ge k+1$ prove the following: $\sum^s_{r=0} {n+r \choose k} = {n+s+1\choose k+1} - {n\choose k+1}$

Question

I found this question online and I can't solve it. I need to solve it without algebraic expansions. Given $s,n,k$ such that $n\ge k+1$ prove the following: $$\sum^s_{r=0} {n+r \choose k} = {n+s+1\choose k+1} - {n\choose k+1}$$

Answer 1

I think it telescopes away by using the recurrence formula for the binomial coefficients: $$\binom{n + r}{k} = \binom{n + r + 1}{k + 1} - \binom{n + r}{k + 1}$$