Let $T_i$ for $i=1,2,...$ be a sequence of i.i.d exponential random variables with common parameter $\lambda$.
Let $N$ be a geometric random variable with parameter $(1/(p+1))$ that is independent of the sequence $T_i$.
Let $X$ be the sum of the $T_i$ from 1 to $N$ Show that the distribution of X is exponential.
I would like to use MGFs. I'm not sure how to incorporate the MGF of N in this case.