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 above-mentioned geometric sequence formulas along with their proofs and examples.

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, ar²... 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.

Here is the list of all geometric sequence formulas. For any geometric sequence a, ar, ar², ar³, ...

• n^th term, a_n = ar^{n - 1} (or) a_n = r a_{n - 1}
• Sum of the first n terms, S_n = a(rⁿ - 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.

We have already seen that a geometric sequence is of the form a, ar, ar², ar³, ...., where 'a' is the first term and 'r' is the common ratio. Here,

• The 1^st term, a₁ = a = a r^1-1
• The 2^nd term, a₂ = a r = a r^2-1
• The 3^rd term, a₃ = a r² = a r^3-1
• .
• .
• .

So in general, the n^th term of a geometric sequence is,

• a_n = ar^n-1

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 a₁₅ of a geometric sequence if a₁₃ = -8 and r = 1/3.

Solution:

By the recursive formula of geometric sequence,

a₁₄ = r a₁₃ = (1/3) (-8) = -8/3

Applying the same formula again,

a₁₅ = r a₁₄ = (1/3) (-8/3) = -8/9.

Therefore, a₁₅ = -8/9.

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, ar², ar³, ... , ar^n-1. Then its sum is denoted by S_n and is given by the formula:

• S_n = a(rⁿ - 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 + ar² + ar³ + ... + ar^n-1... (1)

Multiply both sides by r,

rS_n = ar + ar² + ar³ + ... + arⁿ... (2)

Subtracting (1) from (2),

rS_n - S_n = arⁿ - a

S_n (r - 1) = a (rⁿ - 1)

S_n = a(rⁿ - 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.

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, ar², ar³, ... 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 + ar² + ar³+ ... ...(1)

Multiplying both sides by r,

rS_∞ = ar + ar² + ar³+ ... ... (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.

Here are a few differences between geometric sequence and arithmetic sequence shown in the table below:

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:

• Sequence Calculator
• Series Calculator
• Arithmetic Sequence Calculator
• Geometric Sequence Calculator

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, a₃₄ = a · r³³.

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(rⁿ - 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(rⁿ - 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'.