#! /usr/bin/perl
use warnings;
use strict;
use feature qw{ say };

use List::Util qw{ sum };
for my $A (2..20) {
    for my $n (1..10) {
        die "$A, $n\n"
            unless $A ** $n == 1 + ($A - 1) * sum(map $A ** $_, 0 .. $n - 1)
    }
}

##</code><code>##

\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}