A Geometric Progression (GP) is a sequence of numbers where each term after the first is obtained by multiplying the previous term by a fixed, non-zero number called the common ratio \((r)\). If the first term is \(a\) and the common ratio is \(r\), the GP can be written as: \(a, ar, ar^2, ar^3, … \)
The \(n^{th} \) term of a GP: \(a_n = ar^{n-1} \).
Proof for the Sum of the First n Terms of a GP
If the first term is \(a\) and the common ratio is \(r\), the sum of the first \(n \) terms will be:
\( S_n = a + ar + ar^2 + \dots + ar^{n-1} \quad \text{(1)} \)Multiply both sides by the common ratio \( r \):
\( rS_n = ar + ar^2 + ar^3 + \dots + ar^n \quad \text{(2)} \)Subtract equation (2) from equation (1):
\( \begin{aligned} &S_n – rS_n = a – ar^n \\ &\implies S_n (1 – r) = a(1 – r^n) \\ &\implies S_n = \frac{a(1 – r^n)}{1 – r} \quad \text{(if \( r \neq 1 \))} \\ \end{aligned} \)For \(r=1 \), the GP becomes an arithmetic progression with all terms equal to \( a \), so:
\( S_n = a + a + \dots + a = n a \).
Thus, the sum of the first n terms of a GP can be calculated using the following formula:
\( \displaystyle S_n = \begin{cases} \frac{a(1 – r^n)}{1 – r} & \text{if } r \neq 1, \\ n a & \text{if } r = 1. \end{cases} \)