Sum of Geometric Counts

Let $X_1,\dots,X_r$ be i.i.d. geometric random variables on $\{0,1,2,\dots\}$ with \[ P(X_i=k)=p(1-p)^k. \] Use PGFs to identify the distribution of $S=X_1+\cdots+X_r$ and compute $E[S]$.