# Geometric Series and Positional Notation

By: Gijs Bellaard

A few nights ago I thought of the following cute little proof. Lets start with something easy:
\[ 999 = 1000 - 1\]
Dividing both sides by \(9 = 10 - 1\):
\[ 111 = \frac{1000 - 1}{10-1}\]
So, in general:
\[ 10^n + 10^{n-1} + \dots + 1 = \frac{10^{n+1} - 1}{10-1} \]
But, interestingly, every step we did works in