 Solution

The first term is $ar$, so the finite geometric series is $S=a(r+r^2+\cdots +r^{100})$. Subtracting $rS$ cancels the interior terms and gives $(1-r)S=ar(1-r^{100})$. ThereforeThe rational expression holds for every real $a$ and every real $r\ne 1$, including $r=0$; the same algebra works for complex numbers. No restriction $|r|<1$ is needed for a finite geometric series. The exceptional case follows by adding one hundred equal terms, or by taking the limit as $r\to 1$.