Formula for a Geometric Series
Before knowing the formula for a geometric series, let us recall what is a geometric series. It is a series where the ratios of every two consecutive terms are the same. The formulas for a geometric series include:
 Formula to find the n^{th }term of a geometric sequence.
 Finding the sum of a finite geometric series.
 Finding the sum of an infinite geometric series.
What Is Formula for a Geometric Series?
The formulas for a geometric series include the formulas to find the n^{th} term, the sum of n terms, and the sum of infinite terms. Let us consider a geometric series whose first term is a and common ratio is r.
a + ar + ar^{2} + ar^{3} + ...
n^{th} term = a r^{n1}

The sum formula of a finite geometric series a + ar + ar^{2} + ar^{3} + ... + a r^{n1} is
Sum of n terms = a (1  r^{n}) / (1  r)

The sum formula of an infinite geometric series a + ar + ar^{2} + ar^{3} + ... is
Sum of infinite geometric series = a / (1  r)
Let us learn the applications of the formula for a geometric series in the upcoming sections.
Solved Examples Using Formula for a Geometric Series

Example 1: Find the 10^{th} term of the geometric sequence 1, 4, 16, 64, ...
Solution:
To find: The 10^{th} term of the given geometric sequence.
In the given sequence,
The first term, a = 1.
The common ratio, r = 4 / 1 (or) 16 / 4 (or) 64 / 16 = 4.
Using the formulas of a geometric series, the n^{th} term is found using:
n^{th} term = a r^{n1}
Substitute n = 10, a = 1, and r = 4 in the above formula:
10^{th} term = 1 × 4^{101} = 4^{9} = 262,144
Answer: The 10^{th} term of the given geometric sequence = 262,144.

Example 2: Find the sum of the following geometric series: i) 1 + 1 / 3 + 1 / 9 + ... + 1 / 2187 ii) 1 + 1 / 3 + 1 / 9 + ... using the formula for a geometric series.
Solution:
To find: The sum of the given two geometric series.
In both of the given series,
the first term, a = 1.
The common ratio, r = 1 / 3.
i) In the given series,
n^{th} term = 1 / 2187
a r^{n1} = 1 / 3^{7}
1 × ( 1 / 3) ^{n1} = 3^{7}
3^{n + 1 } = 3^{7}
n + 1 = 7
n = 8
So we need to find the sum of the first 8 terms of the given series.
Using the sum of the finite geometric series formula:
Sum of n terms = a (1  r^{n}) / (1  r)
Sum of 8 terms = 1 ( 1  (1/3)^{8} ) / (1  1/3)
= (1  1 / 6561) / (2 / 3)
= 6560 / 6561 × 3 / 2
= 3280 / 2187
ii) The given series is an infinite geometric series.
Using the sum of the infinite geometric series formula:
Sum of infinite geometric series = a / (1  r)
Sum of the given infinite geometric series
= 1 / (1  (1/3))
= 1 / (2 / 3)
= 3 / 2
Answer: i) Sum = 3280 / 2187 and ii) Sum = 3 / 2