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Geometric Sequence
A geometric sequence is a special type of sequence. It is a sequence in which every term (except the first term) is multiplied by a constant number to get its next term. i.e., To get the next term in the geometric sequence, we have to multiply with a fixed term (known as the common ratio), and to find the preceding term in the sequence, we just have to divide the term by the same common ratio. Here is an example of a geometric sequence is 3, 6, 12, 24, 48, ...... with a common ratio of 2. The common ratio of a geometric sequence can be either negative or positive but it cannot be 0. Here, we learn the following geometric sequence formulas:
 The n^{th} term of a geometric sequence
 The recursive formula of a geometric sequence
 The sum of a finite geometric sequence
 The sum of an infinite geometric sequence
The geometric sequences can be finite or infinite. Here we shall learn more about each of the abovementioned geometric sequence formulas along with their proofs and examples.
What is a Geometric Sequence?
A geometric sequence is a special type of sequence where the ratio of every two successive terms is a constant. This ratio is known as a common ratio of the geometric sequence. In other words, in a geometric sequence, every term is multiplied by a constant which results in its next term. So a geometric sequence is in form a, ar, ar^{2}... where 'a' is the first term and 'r' is the common ratio of the sequence. The common ratio can be either a positive or a negative number.
Geometric Sequence Examples
 1/4, 1/8, .... is a geometric sequence where a = 1/4 and r = 1/2
 4, 2, 1, 1/2, 1/4, ... is a geometric sequence where a = 4 and r = 1/2
 π, 2π, 4π, 8π,.... is a geometric sequence where a = π and r = 2
 √2, √2, √2, √2, ... is a geometric sequence where a = √2 and r = 1
There are two types of geometric sequences based on the number of terms in them. They are
 Finite geometric sequences
 Infinite geometric sequences
Finite geometric sequence
A finite geometric sequence is a geometric sequence that contains a finite number of terms. i.e., its last term is defined. For example 2, 6, 18, 54, ....13122 is a finite geometric sequence where the last term is 13122.
Infinite geometric sequence
An infinite geometric sequence is a geometric sequence that contains an infinite number of terms. i.e., its last term is not defined. For example, 2, −4, 8, −16, ... is an infinite sequence where the last term is not defined.
Geometric Sequence Formulas
Here is the list of all geometric sequence formulas. For any geometric sequence a, ar, ar^{2}, ar^{3}, ...
 n^{th} term, a_{n} = ar^{n  1} (or) a_{n} = r a_{n}_{  1}
 Sum of the first n terms, S_{n} = a(r^{n}  1) / (r  1) when r ≠ 1 and S_{n} = na when r = 1.
 Sum of infinite terms, S_{∞} = a / (1  r) when r < 1 and S_{∞} diverges when r ≥ 1.
Let us study each and every formula one by one here.
n^th Term of Geometric Sequence Formula
We have already seen that a geometric sequence is of the form a, ar, ar^{2}, ar^{3}, ...., where 'a' is the first term and 'r' is the common ratio. Here,
 The 1^{st} term, a_{1} = a = a r^{11}
 The 2^{nd} term, a_{2} = a r = a r^{21}
 The 3^{rd} term, a_{3} = a r^{2 }= a r^{31}
 .
 .
 .
So in general, the n^{th} term of a geometric sequence is,
 a_{n} = ar^{n1}
Here,
 a = first term of the geometric sequence
 r = common ratio of the geometric sequence
 a_{n} = n^{th} term
There is another formula used to find the n^{th} term of a geometric sequence given its previous term and the common ratio which is called the recursive formula of the geometric sequence.
Recursive Formula of Geometric Sequence
We know that in a geometric sequence, a term (a_{n}) is obtained by multiplying its previous term (a_{n}_{  1}) by the common ratio (r). So by the recursive formula of a geometric sequence, the n^{th} term of a geometric sequence is,
 a_{n} = r a_{n}_{  1}
Here,
 a_{n} = n^{th} term
 a_{n}_{  1} = (n  1)^{th} term
 r = common ratio
Example: Find a_{15} of a geometric sequence if a_{1}_{3} = 8 and r = 1/3.
Solution:
By the recursive formula of geometric sequence,
a_{14} = r a_{1}_{3} = (1/3) (8) = 8/3
Applying the same formula again,
a_{15} = r a_{14} = (1/3) (8/3) = 8/9.
Therefore, a_{15} = 8/9.
Sum of Finite Geometric Sequence Formula
The sum of a finite geometric sequence formula is used to find the sum of the first n terms of a geometric sequence. Consider a geometric sequence with n terms whose first term is 'a' and common ratio is 'r'. i.e., a, ar, ar^{2}, ar^{3}, ... , ar^{n1}. Then its sum is denoted by S_{n} and is given by the formula:
 S_{n} = a(r^{n}  1) / (r  1) when r ≠ 1 and S_{n} = na when r = 1.
Let us derive the formula now.
Sum of Finite Geometric Sequence Proof
We have S_{n} = a + ar + ar^{2 }+ ar^{3 }+ ... + ar^{n1}... (1)
Multiply both sides by r,
rS_{n} = ar + ar^{2 }+ ar^{3 }+ ... + ar^{n}... (2)
Subtracting (1) from (2),
rS_{n}  S_{n} = ar^{n } a
S_{n} (r  1) = a (r^{n}  1)
S_{n} = a(r^{n}  1) / (r  1)
Since (r  1) is in its denominator, it is defined only when r ≠ 1. If r = 1, the sequence looks like a, a, a, ... and the sum of the first n terms, in this case, = a + a + a + ... (n times) = na.
Thus, we have derived the formula of the sum of a finite geometric sequence.
Sum of Infinite Geometric Sequence Formula
The sum of an infinite geometric sequence formula gives the sum of all its terms and this formula is applicable only when the absolute value of the common ratio of the geometric sequence is less than 1 (because if the common ratio is greater than or equal to 1, the sum diverges to infinity). i.e., An infinite geometric sequence
 converges (to finite sum) only when r < 1
 diverges (to infinity) when r ≥ 1
Consider an infinite geometric sequence a, ar, ar^{2}, ar^{3}, ... The sum of its infinite terms is denoted by S_{∞}. Then
 S_{∞} = a / (1  r), when r < 1 and S_{∞} diverges when r ≥ 1.
Let us derive the formula now.
Sum of Infinite Geometric Sequence Proof
We have S_{∞} = a + ar + ar^{2 }+ ar^{3}+ ... ...(1)
Multiplying both sides by r,
rS_{∞} = ar + ar^{2 }+ ar^{3}+ ... ... (2)
Subtracting (2) from (1),
S_{∞}  rS_{∞} = a
S_{∞} (1  r) = a
S_{∞} = a / (1  r)
Hence we have derived the formula of the sum of an infinite geometric sequence.
Geometric Sequence vs Arithmetic Sequence
Here are a few differences between geometric sequence and arithmetic sequence shown in the table below:
Geometric Sequence  Arithmetic Sequence 

It is determined by the first term and the common ratio.  It is determined by the first term and the common difference. 
In this, the ratio between every two successive terms is equal to the same number.  In this, the difference of every two successive terms is equal to the same number. 
The terms form an exponential relation.  The terms form a linear relation. 
The sum of an infinite geometric sequence may converge or diverge.  The sum of an infinite arithmetic sequence always diverges. 
Important Notes on Geometric Sequence:
 In a geometric sequence, every term is obtained by multiplying its previous term by a constant (r, which is called the common ratio).
 If r > 1, then the terms are in ascending order.
 When r = 1, and the first term of a geometric sequence is 'a' then its sum is a · n.
 When r ≥ 1, the infinite geometric sequence diverges.
☛ Related Topics:
Geometric Sequence Examples

Example 1: Find the 25^{th} term of the geometric sequence 1, 1/5, 1/25, 1/125, ...
Solution:
Here, the first term is, a = 1 and
the common ratio is, r = (1/5) / 1 = (1/25) / (1/5) = ... = 1/5.
The n^{th} term of a geometric sequence is calculated using,
a_{n} = ar^{n  1}
Substituting n = 25 here,
a_{25} = (1) (1/5)^{251} = 1/5^{24}
Answer: The 25^{th} term of the given geometric sequence is 1/5^{24}.

Example 2: Find the sum of the first 18 terms of the geometric sequence 2, 6, 18, 54, .....
Solution:
Here, the first term is, a = 2.
The common ratio, r = 6/2 = 18/6 = 54/18 = ... = 3.
Number of terms is, n = 18.
The sum of finite geometric sequence formula is,
S_{n} = a(r^{n}  1) / (r  1)
S_{1}₈ = 2 (3^{18}  1) / (3  1) = 3^{18}  1.
Answer: The sum of the first 18 terms of the given geometric sequence is 3^{18}  1.

Example 3: Find the following sum of the terms of this infinite geometric sequence: 1/2, 1/4, 1/8... ∞
Solution:
Here, the first term is, a = 1/2
Common ratio is, r = (1/4) / (1/2) = (1/8) / (1/4) = ... = 1/2
We have r = 1/2 = 1/2 < 1. So the geometric sequence converges.
So the sum of infinite geometric sequence is,
S_{∞} = a/(1r)
S_{∞} = (1/2)(1  1/2) = (1/2) / (1/2) = 1
Answer: The sum of the given infinite geometric sequence is 1.
FAQs on Geometric Sequence
What is the Definition of Geometric Sequence?
A geometric sequence is a sequence of numbers in which the ratio of every two successive terms is the constant. This constant is called the common ratio of the geometric sequence.
How to Find the n^{th} Term of Geometric Sequence?
If 'a' is the first term and 'r' is the common ratio of a geometric sequence, then its n^{th} term is calculated using the formula, a_{n} = a · r^{n  1}. For example, a_{34} = a · r^{33}.
What is r in Geometric Sequence Formula?
'r' in any geometric sequence formula represents the common ratio. It is the ratio of any term of the geometric sequence to its previous term. For example, the common ratio of the geometric sequence 1, 2, 4, 8, ... is 2.
When Does a Geometric Sequence Diverge?
A geometric sequence with the common ratio 'r' diverges when r ≥ 1. It converges only when r < 1. Here r is the absolute value of 'r'.
What is the Meaning of Common Ratio of Geometric Sequence?
In a geometric sequence, every term is obtained by multiplying its previous term by a fixed number. That fixed number is called the common ratio of the geometric sequence.
How to Find the Sum of a Finite Geometric Sequence?
The sum of the first n terms of a geometric sequence with first term 'a' and common ratio 'r' is calculated using the formula S_{n} = a(r^{n}  1) / (r  1) when r ≠ 1 and S_{n} = na when r = 1.
What Are Geometric Sequence Formulas?
Here are the geometric sequence formulas where 'a' is the first term and 'r' is the common ratio.
 a_{n} = ar^{n  1}
 S_{n} = a(r^{n}  1) / (r  1), when r ≠ 1
 S_{∞} = a / (1  r) when r < 1
How to Find the Sum of an Infinite Geometric Sequence?
The sum of an infinite geometric sequence with the first term 'a' and common ratio 'r' is calculated using the formula S = a / (1  r) when r < 1. If r ≥ 1, then the sum diverges (to infinity).
When Does a Geometric Sequence Converge?
A geometric sequence with a common ratio 'r' converges when r is less than 1. It diverges only when r ≥ 1, where r is the absolute value of 'r'.
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