Geometric Sequence
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Last updated on December 15, 2025

A geometric sequence is calculated by multiplying the previous one by the same fixed number, known as the common ratio. This kind of sequence is used in areas like mathematics, science, finance, and computer simulations to model situations involving exponential increase.

What is a Geometric Sequence

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A geometric sequence is a list of numbers, where each term is obtained by multiplying the previous term by a constant called the common ratio. If r > 1, the sequence grows; if 0 < r < 1, it decreases. Real-life examples include population growth or scientific experiments.

Examples of Geometric Sequences
 

  • 2, 4, 8, 16, 32, …. These are increasing geometric sequences. The first term a is 2, and the common ratio, r, is 2. Each term is double the previous one. 
     
  • 100, 50, 25, 12.5, … in this sequence, a = 100 and r = \(\frac{1}{2}\). Each term is half of the previous one; hence, it is a decreasing geometric sequence. 
     
  • 3, -6, 12, -24, … This is an example of an alternating geometric sequence. Here, the common ratio r = -2, which causes the signs of each term to alternate. 

Based on the number of terms a sequence has, geometric sequences are of two types. They are: 
 

  • Finite geometric sequences
     
  • Infinite geometric sequences. 
     

Finite geometric sequence

A finite geometric sequence has a limited number of terms. It has a clear beginning and end. The geometric sequence 5, 10, 20, 40 is an example of a finite geometric sequence. 

Infinite geometric sequence 

An infinite geometric sequence has an infinite number of terms and continues indefinitely. For example, 1, \(\frac{1}{2}\), \(\frac{1}{4}\), \(\frac{1}{4}\), …….

Geometric Sequence Formulas

A geometric sequence is a chain of numbers in which every term is obtained by multiplying the previous term by a fixed number, known as the common ratio.

1. nth Term of a Geometric Sequence

 \(a_n = a_1 \times r^{\,n-1}\)

Where, 

an= the nth term
‘
a = first term


r = common ratio


n = term number

2. Sum of the First n Terms (Finite Sum)

\(S_n = \frac{a_1 (1 - r^n)}{1 - r}, \quad \text{for } r \neq 1\)


Where, 
 

\(S_n\)​ = sum of first n terms

a = first term

r = common ratio

3. Sum to Infinity (Infinite Geometric Series)

\(S_\infty = \frac{a_1}{1 - r}, \quad \text{for } |r| < 1\)

Where, 

S∞​ = infinite sum 


a = first term


r = common ratio (must be between 1< r < 1)

nth Term of a Geometric Sequence Formula

A geometric sequence is calculated by multiplying the previous one by a fixed number, known as the common ratio. The nth term formula is:

\(a_n = a_1 \times r^{\,n-1}\)

Where:

an = the nth term


a = the first term of the sequence


r  = the common ratio


n = the position of the term in the sequence

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Recursive Formula of Geometric Sequence

A recursive formula says that every term in a sequence is based on its preceding term(s). In a geometric sequence, every term is obtained by multiplying the previous term by a fixed number known as the common ratio. The recursive formula is:

\(a_n = r \times a_{n-1}, \quad \text{for } n \ge 2\)

Where:

an​ = the nth term

an-1 ​ = the previous term

r  = the common ratio

You must also specify the first term: a1​

Example: If a1 = 2 and r = 3, then the sequence is:

2, 6, 18, 54, …

Recursive formula:

a1 = 2

\(a_n = a_{n-1} \times 0.3, \quad \text{for } n > 1\)

Sum of Finite Geometric Sequence Formula

The formula for the sum Sn ​of the first n terms of a finite geometric sequence is:

\(S_n = \frac{a(1 - r^n)}{1 - r}, \quad \text{for } r \neq 1\)

Where:

a1 is the first term,


r is the common ratio between consecutive terms?


n is the number of terms to sum?

If the common ratio r = 1, the series becomes a constant sequence, and the sum is simply:

Sn= n × a

Derivation

To derive this formula, consider the geometric series:

\(S_n = a + a r + a r^2 + a r^3 + \dots + a r^{\,n-1} \)

Multiply both sides by the common ratio (r):

\(r S_n = a r + a r^2 + a r^3 + \dots + a r^n\)

Subtract the original series from this new equation:

\(r S_n - S_n = \bigl(a r + a r^2 + a r^3 + \dots + a r^n \bigr) - \bigl(a + a r + a r^2 + \dots + a r^{\,n-1} \bigr)\)

Simplifying the right-hand side:

\((r - 1) S_n = a r^n - a\)

Solving for Sn 

\(S_n = \frac{a(1 - r^n)}{1 - r}, \quad \text{for } r \neq 1\)

Example: Consider a geometric series with the first term a = 4, the common ratio r = 3, and n = 6 terms.

Given: First term a = 4

Common ratio r = 3

Number of terms n = 6

Using the formula: \(S_6 = 4 \times \frac{1 - 3^6}{1 - 3}\)

First, calculate 36:

\(3^6 = 729\)

Now, we will substitute into the formula:

\(S_6 = 4 \times \frac{1 - 729}{1 - 3} = 4 \times \frac{-728}{-2}\)

\(S_6 = 4 \times 364 = 1456\)

The answer is 1456.

Formula for the Sum of an Infinite Geometric Sequence

An infinite geometric series is a sum of terms, where each term is multiplied by the same common ratio (r) to get the next one.

The sum exists only if the absolute value of the ratio is less than 1 (|r| < 1).

Formula: \(S_\infty = \frac{a}{1 - r}, \quad \text{for } |r| < 1\)

Where:

a = first term

r = common ratio

If |r| ≥ 1, the series doesn’t have a finite sum (it diverges).

Example:

Series:\( 2 + 1 +\frac{1}{2} + \frac{1}{4} + \frac{1}{3} + …\)

a = 2

r = \(\frac{1}{2}\)

\(S_\infty = \frac{2}{1 - ½} = \frac{2}{½} = 4\)

The sum of the series is 4.

Difference Between Geometric Sequence and Arithmetic Sequence

Arithmetic Sequence

Geometric Sequence

An arithmetic sequence is a sequence in which each term is formed by adding a fixed number to the previous term. 

A geometric sequence is a sequence in which each term is formed by multiplying the previous term by a fixed non-zero number.

The fixed number is known as constant common difference (d). 

The fixed number is known as common ratio (r).

The same amount is added or subtracted each time. 

Each term is multiplied or divided by the same value each time. 

Formula for the nth term :
\(a_n = a + (n - 1)d\)

Formula for the nth term:
 \(a_n = a_1 × r ^{n-1}\) 

It follows a linear growth, where it has constant increase or decrease. 

It follows exponential growth or decay, which means rapid increase or decrease.

Negative change will occur on arithmetic sequence, when the common difference is negative. 

Negative change will occur on geometric sequence, when the common ratio is negative. 

Example for arithmetic sequence are: 3, 6, 9, 12, 15.

Example for geometric sequence are: 2, 4, 8, 16, 32.

Tips and Tricks for Geometric Sequence

Geometric sequences are number patterns where each term is obtained by multiplying the previous term by a constant ratio. With the right tips and tricks, you can quickly find terms, calculate sums, and solve problems efficiently without getting lost in lengthy calculations.

  • Always remember that negative ratios produce alternating sequences. Use absolute value to track magnitude, handle sign separately. Fractional ratios decrease the sequence toward 0. Useful in finance (depreciation, discounting).
  • Many sequences use powers of numbers (2, 3, 10, etc.), which can help you calculate terms faster without a calculator.
  • When n is large, express terms in powers rather than expanding them.
  • If the common ratio is negative, the sequence alternates signs. Track the magnitude and sign separately to avoid errors.
  • Divide any term by its previous term to find the common ratio. This helps you spot the pattern fast and avoid mistakes.
  • Parents and teachers should encourage students to identify the first term (a) and the common ratio (r) before solving geometric sequence problems. This will give them clarity and prevent them from getting confused with arithmetic sequences. 
  • Use real-life contexts such as population growth, compound interest, or so on, to help students understand why geometric sequences increase or decrease rapidly. 
  • Parents and teachers can demonstrate geometric sequences by repeatedly folding paper or doubling objects, making the concept more visual and engaging for learners. 
  • Encourage students to compare two consecutive terms by division to verify the common ratio. This will help them confirm whether a sequence is geometric. 
  • Have students estimate the sizes of terms in a geometric sequence before calculating them. It will build their number sense and help with easy calculation.

Common Mistakes on Geometric Sequences and How to Avoid Them

When learning geometric sequences, students often make small errors that can lead to wrong answers. Understanding these common mistakes, and knowing how to avoid them, can help you solve problems more confidently and accurately. 

Real-Life Applications of Geometric Sequences

A geometric sequence is a pattern where each term is obtained by multiplying the previous term by a constant ratio. This concept is not just theoretical—it appears in many real-world scenarios:

Robotics: The lengths or angles of robotic arm segments can increase geometrically to reach higher points efficiently, with each segment following a fixed ratio for smooth motion.

Architecture: Architects often use geometric sequences when designing structures like spiral staircases or tiered platforms. The height and width of each step or level can follow a geometric pattern to ensure balance and aesthetics.

Art and Design: Artists use geometric sequences to create perspective and depth in drawings or designs. Objects may decrease in size geometrically as they appear further away, creating realistic visual effects.

Finance: Geometric sequences appear in compound interest calculations. Each term represents the total amount after successive periods, where the amount grows by a fixed ratio each time.

Engineering: The sizes of structural elements, like steps in a staircase or layers in a tower, can follow a geometric sequence to maintain balance, safety, and proper load distribution.

Problem 1

What is the 5th term of the sequence: 3, 6, 12, 24, ...?

Okay, lets begin

The 5th term is 48.

Explanation

First term (a₁): 3

Common ratio (r): \(6 ÷ 3 = 2\)

Now we will use the formula for the nth term of a geometric sequence.

\(a_n = a_1 \times r^{\,n-1}\)

For the 5th term:

\(a_5 = 3 \times 2^{5-1} = 3 \times 2^4 = 3 \times 16 = 48\)

Well explained 👍

Problem 2

Find the 8th term of the sequence: 2, 6, 18, 54, ...

Okay, lets begin

The 8th term is 4374.

Explanation

First term (a₁): 2

Common ratio (r): 6 ÷ 2 = 3. Using the nth term formula:

\(a_8 = 2 \times 3^{8-1} = 2 \times 3^7 = 2 \times 2187 = 4374\)

Well explained 👍

Problem 3

Determine the 6th term of the sequence: 2, 6, 18, 54, ...

Okay, lets begin

The 6th term is 486.

Explanation

First term (a₁): 2

Common ratio (r): \(6 ÷ 2 = 3 \)
Using the nth term formula:

\(a_6 = 2 \times 3^{6-1} = 2 \times 3^5 = 2 \times 243 = 486\)

Well explained 👍

Problem 4

What is the next term in the sequence: 48, 24, 12, 6, ...?

Okay, lets begin

The next term is 3.

Explanation

First term (a₁): 48

Common ratio (r):\( 24 ÷ 48 = 0.5\)


Using the nth term formula:

\(a_5 = 48 \times 0.55 - 1 = 48 \times 0.54 = 48 \times 0.0625 = 3\)

Well explained 👍

Problem 5

The first term of a geometric sequence is 𝑎 = 5 and the common ratio is 𝑟 = − 2 . Find the sum of the first 5 terms.

Okay, lets begin

S5 = 55

Explanation

\(S_n = a \frac{1 - r^n}{1 - r} \text{Substitute the values:}\)

\( \\ S_5 = 5 \frac{1 - (-2)^5}{1 - (-2)} = 55\)

\(5 \frac{1 - (-32)}{1 + 2} = 5 \frac{1 + 32}{3} = 5 \times \frac{33}{3} = 5 \times 11\)

\(= 55\)

Well explained 👍

FAQs of Geometric Sequence

1.What is a geometric sequence?

A geometric sequence is a sequence of numbers in which every term is obtained by multiplying the previous term by a constant value known as the common ratio.

2.How do you find the common ratio?

To find the common ratio (r) in a geometric sequence, divide any term by its immediate predecessor. The formula is:

r= an/an-1

  •  An is the nth term,
  • an-1th Term.

For example, in the sequence 3, 9, 27, 81, 243, ..., the common ratio is:

r=9/3=3

3.Can a geometric sequence have a common ratio of 0?

No, a geometric sequence cannot have a common ratio of 0. If r = 0, all terms after the first term become 0, for example, x, 0, 0, 0, ….

4.Can a geometric sequence have a negative common ratio?

Yes, if the common ratio is negative, the terms will alternate between positive and negative.

5.Can a geometric sequence have fractions or decimals?

Yes, a geometric sequence can have fractions or decimals as terms. Both the first term and the common ratio can be fractions or decimals, and the sequence will still follow the geometric progression rules.

6.Why should parents help their child understand geometric sequences?

Learning geometric sequences helps children spot patterns, improve problem-solving skills, and understand concepts like growth and scaling, which are useful in everyday life, finance, and science.

7.How can geometric sequences prepare a child for higher math?

Builds foundation for algebra and exponential functions. Helps understand growth patterns, like interest rates, population growth, or computer algorithms. Strengthens logical thinking and pattern recognition skills.

Jaskaran Singh Saluja

About the Author

Jaskaran Singh Saluja is a math wizard with nearly three years of experience as a math teacher. His expertise is in algebra, so he can make algebra classes interesting by turning tricky equations into simple puzzles.

Fun Fact

: He loves to play the quiz with kids through algebra to make kids love it.