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Geometric Series Formula
Before knowing the geometric series formula, let us recall what is a geometric series. It is a series (sum of terms) where the ratios of every two consecutive terms are the same. The geometric series formula includes:
 Formula to find the n^{th }term of a geometric series.
 Finding the sum of a finite geometric series.
 Finding the sum of an infinite geometric series.
What is a Geometric Series?
A geometric series is the sum of finite or infinite terms of a geometric sequence. For the geometric sequence a, ar, ar^{2}, ..., ar^{n1}, ..., the corresponding geometric series is a + ar + ar^{2 }+ ..., ar^{n1 }+ .... We know that "series" means "sum". In particular, the geometric series means the sum of the terms that have a common ratio between every adjacent two of them. There can be two types of geometric series: finite and infinite. Here are some examples of geometric series.
 1/2 + 1/4 + .... + 1/8192 is a finite geometric series
where the first tern, a = 1/2 and the common ratio, r = 1/2  4 + 2  1 + 1/2  1/4 + ... is an infinite geometric series
where the first term, a = 4 and the common ratio r = 1/2
Geometric Series Formula
The geometric series formula refers to the formula that gives the sum of a finite geometric sequence, the sum of an infinite geometric series, and the n^{th} term of a geometric sequence. The sequence is of the form {a, ar, ar^{2}, ar^{3}, …….} where, a is the first term, and r is the "common ratio".
Geometric Series Formulas
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} + ...
Formula 1: The n^{th} term of a geometric sequence is,
n^{th} term = a r^{n1}
Where,
 a is the first term
 r is the common ratio of every two successive terms
 n is the number of terms.
To see how this formula is derived, click here.
Formula 2: 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) (or) a (r^{n}  1) / (r  1)
where,
 a is the first term
 r is the common ratio every two consecutive terms
 n is the number of terms.
To see how this formula is derived, click here.
Formula 3: The sum formula of an infinite geometric series a + ar + ar^{2} + ar^{3} + ... is
Sum of infinite geometric series = a / (1  r)
where,
 a is the first term
 r is the common ratio every two successive terms
To see how this formula is derived, click here.
Convergence of Geometric Series
A finite geometric series always converges. But the convergence of an infinite geometric series depends upon the value of its common ratio. An infinite geometric series a, ar, ar^{2}, ...
 converges when r < 1 and hence we can find its sum using the formula a / (1  r).
 diverges when r > 1 and hence we can't find its sum in this case.
Examples:
The geometric series 2 + 4 + 8 + 16 + .... diverges as r = 2 = 2 > 1. So its sum can't be found.
The geometric series 1  1/2 + 1/4  1/8 + 1/16  .... converges as r = 1/2 = 1/2 < 1. Hence, its sum is, a / (1  r) = 1 / (1  (1/2)) = 1/ (3/2) = 2/3.
Examples Using Formula for a Geometric Series

Example 1: Find the 10^{th} term of the geometric series 1 + 4 + 16 + 64 + ...
Solution:
To find: The 10^{th} term of the given geometric series.
In the given series,
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
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

Example 3: Calculate the sum of the finite geometric series if a = 5, r = 1.5 and n = 10.
Solution:
To find: the sum of geometric series
Given: a = 5, r = 1.5, n = 10
s_{n} = a(1−r^{n})/(1−r)
The sum of ten terms is given by S_{10 }= 5(1−(1.5)^{10})/(1−1.5)
= 566.65
Answer: The sum of the geometric series is 566.65.
FAQs on Geometric Series Formula
What is the Difference Between Geometric Series and Geometric Sequence?
A geometric sequence is the collection of terms where the ratio of every two successive terms is the same whereas a geometric series is the "sum" of the terms of the geometric sequence.
What Are the Geometric Series Formulas in Math?
The geometric series formulas are the formulas that help to calculate the sum of a finite geometric sequence, the sum of an infinite geometric series, and the n^{th} term of a geometric sequence. These formulas are geometric series with first term 'a' and common ratio 'r' given as,
 n^{th} term = a r^{n1}
 Sum of n terms = a (1  r^{n}) / (1  r)
 Sum of infinite geometric series = a / (1  r)
What is the Infinite Geometric Series Formula?
The sum formula of an infinite geometric series a + ar + ar^{2} + ar^{3} + ... can be calculated using the formula, Sum of infinite geometric series = a / (1  r), where a is the first term, r is the common ratio for all the terms and n is the number of terms.
Is it possible to Find the Sum of the Geometric Series Always?
If the geometric series is finite, we can find its sum always. But if it is infinite, then its sum can be found only when the absolute value of its common ratio is less than 1.
When Does a Geometric Sequence Converge?
We know that 'r' represents the common ratio between every two consecutive terms of a geometric series. An infinite geometric series converges only when r < 1 and we can find its sum only if it converges.
How To Use the Geometric Series Formula?
To use the geometric series formula
 Step 1: Check for the given values, a, r and n.
 Step 2: Put the values in the geometric series formula as per the requirement  the sum of a finite geometric sequence, the sum of an infinite geometric series, or the n^{th} term of a geometric sequence
What are the Applications of Geometric Series?
Geometric series formulas are used throughout mathematics. These have important applications in physics, engineering, biology, economics, computer science, queueing theory, and finance.
What is 'r' in the Geometric Series Formula?
In the geometric series formula, 'r' refers to the common ratio. The formulas for geometric series with 'n' terms and the first term 'a' are given as,
 Formula for nth term: n^{th} term = a r^{n1}
 Sum of n terms = a (1  r^{n}) / (1  r)
 Sum of infinite geometric series = a / (1  r)
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