Combo Identity: How to prove this using Induction 
$$ \sum_{n = 0}^{\infty} \binom{n + k}{k}x^n = \dfrac{1}{(1 - x)^{k + 1}} $$

Could someone suggest how I should get started to prove this using induction?

You start like that:
For $k=0$: $$\sum_{n=0}^{\infty} \binom{n}{n} x^n=\sum_{n=0}^{\infty} x^n=\frac{1}{1-x} \checkmark$$
Then,we suppose that: $$\sum_{n=0}^{\infty} \binom{n+k}{k} x^n=\frac{1}{(1-x)^{k+1}}$$
For $k+1$:
$$\sum_{n=0}^{\infty} \binom{n+k+1}{k+1} x^n=\sum_{n=0}^{\infty} \frac{n+k+1}{k+1} \binom{n+k}{k}x^n$$
then you have to continue,in order to find a relation that stands.
