Sum of simplicial Polytopic numbers

Question

Is the following assertion ,I have made, true? If so, how to prove it:

$$ \sum_{n=1}^{k}\frac{n(n+1)(n+2)\cdots(n+r)}{(r+1)!}=\frac{k(k+1)(k+2)\cdots(k+r)(k+r+1)}{(r+2)!}$$

Answer 1

$$\sum_{n=1}^{k}\frac{(n+r)!}{(n-1)!(r+1)!} = \sum_{n=1}^{k}\binom{n+r}{r+1} = \binom{k+r+1}{r+2} $$ is a straightforward consequence of the hockey-stick identity.