# Arithmetic Progression \begin{array}{ccccccccccc} S & = & a & + & (a + d) & + & (a + 2d) & + & \dots & + & b & \quad(n \text{ terms})\\ S & = & b & + & (b - d) & + & (b - 2d) & + & \dots & + & a & \quad(n \text{ terms})\\ \hline 2S & = & (a + b) & + & (a + b) & + & (a + b) & + & \dots & + & (a + b) & \quad(n \text{ terms}) \end{array} Thus \( 2S = n(a + b) \) or \[ S = \frac{n(a + b)}{2}. \] ## Example \[ 1 + 2 + \dots + n = \frac{n(n + 1)}{2}. \]