sequences and series - Proof by induction of $sum_{n=1}^{infty }left ( x+1 right )^{-n}=frac{1}{x}$

I've been trying to work on some proof for class and I basically want to prove that:

$$\sum_{n=1}^{\infty }\left ( x+1 \right )^{-n}=\frac{1}{x},\quad \text{where} \quad x \in \mathbb{Z^{+}}.$$

So far I've been trying to use proof by induction, but I can't seem to get anywhere as it has no final term. Does anyone have any idea how I could go about proving this?

Answer

Guide:

Use geometric series.

Notice that if $x \in \mathbb{Z}^+$, then $0<\frac{1}{x+1} < 1$.