# Geometric Progression \begin{array}{ccccccccccc} S & = & a & + & ar & + & ar^2 & + & \dots & + & b & \quad(n \text{ terms}) \\ \\ -S & = & -a & - & ar & - & ar^2 & - & \dots & - & b & \quad(n \text{ terms})\\ rS & = & ar & + & ar^2 & + & ar^3 & + & \dots & + & br & \quad(n \text{ terms}) \\ \hline S(r - 1) & = & br - a \end{array} Thus \[ S = \frac{br - a}{r - 1}. \] Since \( b = ar^{n - 1} \) we get \[ S = \frac{a(r^n - 1)}{r - 1}. \] # Example \[ 1 + 2 + 4 + \dots + 2^{n-1} = 2^n - 1. \]