Sum of Geometric Progression

Geometric progression is defined as series formed by multiplication of a constant term to obtain next term of series. Here this constant term is as common ratio. Ratio of two consecutive terms in a Geometric Sequence is constant which is defined as common Ratio and represented by 'r'. Sum of Geometric Progression is termed as Geometric Series. This series includes terms in a geometric progression. As we know that progression refers to increment or process of any identity in any particular format or in any specified pattern. sum of geometric progression involves sum of finite terms of series as well as sum of infinite terms. The n th term of a geometric series is determined by the expression a_n = ar ^{n – 1} where 'a' is first or initial term and 'r' is common ratio between two terms of series. The series which follows a recursive relation is represented as a_n = r a^{n – 1}. It is defined for Integer whose value is greater than or equal to 1. Sum of a geometric progression when series start from initial value is given as:
S_n = a (1 – rⁿ⁺¹) / (1 - r).
And when initial value does not starts from zero that is it has some value for initial term which is represented as:
S_n = a (r^m – rⁿ⁺¹) / (1 - r) where 'm' is degree of initial term when it is not zero.
Sum of infinite series of a geometric progression is represented as
S_n = a / (1 - r).
For example: Consider a series 2, 4, 8, 16. . . . . . . . . . up to infinity. Here, initial term or first term ‘a’ is equals to 2 and common ratio that is ‘r’ is equals to 2. Then sum of this series is determined by
S_∞ = a / (1 - r).