\documentclass{article}
\title{Summation Fulguration}
\newtheorem{thm}{Theorem}
\newtheorem{prf}{Proof}[thm]
\begin{document}
\begin{thm}
$$a^n = 1 + (a - 1) \sum_{i=0}^{n-1} a^i$$
\end{thm}
\begin{prf}
$$1 + (a - 1) \sum_{i=0}^{n-1} a^i$$
$$= 1 + \sum_{i=0}^{n-1}(a-1)a^i$$
$$= 1 + \sum_{i=0}^{n-1} a^{i+1} - a^i$$
$$= 1 + \sum_{i=0}^{n-1}a^{i+1} - \sum_{i=0}^{n-1}a^i$$
$$= 1 + (\sum_{i=1}^{n-1}a^i) + a^n - (a^0 + \sum_{i=1}^{n-1}a^i)$$
$$= 1 + (\sum_{i=1}^{n-1}a^i) + a^n - a^0 - \sum_{i=1}^{n-1}a^i$$
$$= 1 + a^n - a^0$$
$$= 1 + a^n - 1$$
$$= a^n$$
Q.E.D.
\end{prf}
\end{document}