\begin{equation*} \sum_{x=r}^{\infty} {\binom{x - 1}{r-1}(1-p)^{x-r}p^r} = p^r \sum_{x=r}^{\infty} {\binom{x - 1}{r-1}(1-p)^{x-r}} \end{equation*}

\begin{align*} & = p^r \sum_{k=0}^{\infty} {\binom{r + k - 1}{k}(1-p)^k}\\ & = p^r \frac{1}{(1-(1-p))^r}\\ & = 1 \end{align*}