Geometric Sum Formula
Before going learn the geometric sum formula, let us recall what is a geometric sequence. A geometric sequence is a sequence where every term has a constant ratio to its preceding term. A geometric sequence with the first term a and the common ratio r and has a finite number of terms is commonly represented as a, ar, ar^{2}, ..., ar^{n1}. A geometric sum is the sum of the terms in the geometric sequence. The geometric sum formula is used to calculate the sum of the terms in the geometric sequence.
What Is the Geometric Sum Formula?
There are two geometric sum formulas. One is used to find the sum of the first n terms of a geometric sequence whereas the other is used to find the sum of an infinite geometric sequence.
The geometric sum formula for finite terms is given as:
If r = 1, S_{n} = na
If \(r \neq 1, \, S_{n}=\dfrac{a(1r^{n})}{1r}\)
The geometric sum formula for infinite terms is given as:
\(S_{n}=\dfrac{a}{1r}\)
Where
 a is the first term
 r is the common ratio
 n is the number of terms
Let us see the applications of the geometric sum formulas in the following section.
Solved Examples Using Geometric Sum Formula

Example 1: Find the sum of the terms 1/3 + 1/9 + 1/27 + ... to ∞ using the geometric sum formula?
Solution:
To find: geometric sum
Given:
a = 1/3, r = 1/3
Using geometric sum formula for infinite terms,
S_{n} = a /(1r)
S_{n} = (1/3)( 1  1/3)
S_{n} = 1/2
Answer: Geometric sum of the given terms is 1/2.

Example 2: Calculate the sum of series 1/5, 1/5, 1/5, .... if the series contains 34 terms.
Solution:
To find: geometric sum
Given:
a = 1/5, r = 1, and n = 34
Using geometric sum formula for finite terms,
S_{n} = na
S_{n} = 34 × 1/5
S_{n} = 6.8
Answer: Geometric sum of the given terms is 6.8.