Geometric series formula

We now derive the sum of the first \(n\) terms of a G.P. The method used is very different from that used for deriving the sum of the first \(n\) terms of an A.P.

Consider the sum of the first \(n\) terms of a G.P., whose first term is \(a\) and common ratio \(r\), denoted by \(S_n\).

\[S_n = a + ar + ar^2 + \dots + ar^{n-1}\]

Multiplying this equation throughout by \(r\), we have

\[rS_n = ar + ar^2 + ar^3 + \dots + ar^n\]

Subtracting the first equation from the second:

\[ \begin{align*} rS_n - S_n & = (ar + ar^2 + ar^3 + \dots + ar^{n-1} + ar^n) \\ & - (a + ar + ar^2 + \dots + ar^{n-1}) \\ (r - 1) S_n &= ar^n - a \\ S_n &= \frac{ar^n - a}{r - 1} \\ S_n & = \frac{a(r^n - 1)}{r - 1}. \end{align*} \]

So, the sum \(S_n\) of the first \(n\) terms of a G.P. is given by

\[S_n = \frac{a(r^n - 1)}{r - 1}, \quad r \neq 1\]