A geometric series is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. For example, in the series 2, 6, 18, 54, each term is multiplied by 3, which is the common ratio. The sum of a geometric series can be calculated using a specific formula, especially when the series has a finite number of terms. The formula for the sum S_n of the first n terms is S_n = a \frac(1 - r^n)(1 - r) , where a is the first term and r is the common ratio.