Summation Of Series of $\binom{x+k}{k+1}$ where $k$ is $0$ to $n$

Question

Want The formula or to find The Sum of Series where

$$S=\sum_{k=0}^n \binom{x+k}{k+1}$$

where $x$ is any constant $\geq 1$ and $n$ is another constant.

Answer 1

By Pascal's identity,

$$\binom{x+k}{k+1} = \binom{x+k+1}{k+1} - \binom{x+k}{k}.$$

So by telescoping,

$$S = \binom{x+n+1}{n+1} - \binom{x}{0} = \binom{x+n+1}{n+1}- 1.$$

Answer 2

$$\begin{align} \sum_{k=0}^n \binom{x+k}{k+1} &=\sum_{k=0}^n \binom{x+k}{x-1}\\ &=\sum_{k=0}^n \left[\binom{x+k+1}{x}-\binom{x+k}x\right]\\ &=\sum_{k=1}^{n+1} \binom{x+k}x-\sum_{k=0}^{n}\binom{x+k}x\\ &=\binom{x+n+1}x-1\qquad\blacksquare \\ \end{align}$$