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?

Why do you need proof by induction here? Use the Binomial Theorem! Simply set $-k-1 = v$:
$$
(1-x)^{v} = \sum_{j=0}^{\infty}\binom{v}{j}(-x)^j
$$
Now use the property of the binomial coefficient: $\binom{v}{j} = v (v-1) \ldots (v-j+1)\cdot \frac{1}{j!}$ Now use the definition for $v$: $\binom{-k-1}{j} = (-1)^j \frac{(k+1)(k+2) \ldots (k+j-1)}{j!} = (-1)^j =\frac{(k+j-1)!}{k! j!} = \frac{
(-1)^j}{k} \binom{k+j-1}{j}$. The $(-1)^j$ term cancels out, and the expression becomes what you wanted to prove.
