Solution
= Solution
For fixed $n\geq0$, use <mathematical induction> on $k$. At $k=0$, both sides equal one. If the <hockey-stick identity> holds at $k$, then <Pascal's identity> gives
$$
\sum_{i=0}^{k+1}\binom{n+i}i
=\binom{n+k+1}k+\binom{n+k+1}{k+1}
=\binom{n+k+2}{k+1}.
$$
This is the required formula with $k$ replaced by $k+1$.