Prove that $S_N = \sum_{i=1}^{N} X_{i}$ is exponentially distributed with parameter $p$

Question

Let $X_{1}, X_{2}, \ldots$ be exponential random variables with parameter $1 .$

Let $N$ be a geometric random variable with parameter $p$. Prove that $S_N = \sum_{i=1}^{N} X_{i}$ is exponentially distributed with parameter $p$.

The sum of n exponential random variables with parameter $β$ is a $gamma (n, β)$ random variable, therefore $f_{S_N}(s) = \frac{1}{(N-1)!}s^{N-1}e^{-s}$ and it should be equal to $pe^{-ps}$. I just don't see how this can be possible. I feel like I have to apply Poisson process with rate 1 as $X_i$'s are interarrival times and $f_{S_N}(s) = P_{N-1}(s)$.

Any hints?