Geometric Progression
2026-02-28 17:28 Diff
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Last updated on October 18, 2025

ChatGPT said: A Geometric Progression (GP) is a sequence where each term is obtained by multiplying the previous term by a fixed common ratio. The previous term can be found by dividing by this ratio. For example, 3, 6, 12, 24,… is a GP with a ratio of 2. GPs can have finite or infinite terms, and this article covers their meaning, formulas, and types.

What is a Geometric Progression?

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The sequence in which each term is obtained by multiplying the previous term by a fixed number (common ratio) is known as a geometric progression. It is usually expressed as: \(a, ar, ar^2, ar^3…\), where ‘a’ represents the first term and ‘r’ represents the common ratio. The common ratio can be positive or negative. Any term in a GP can be determined using the first term and the common ratio.

What are the Types of Geometric Progression?

Geometric progressions are mainly classified into two types based on their length.

The different types of geometric progressions are:

  • Finite geometric progression
     
  • Infinite geometric progression


We will now learn about each type in detail:

Finite geometric progression: A finite geometric progression has a limited number of terms, and the last term is known. For example: \(\frac{1}{2}\), \(\frac{1}{4}\), \(\frac{1}{8}\), \(\frac{1}{16}\), …, \(\frac{1}{32768}\) is a finite geometric progression. Here, \(\frac{1}{32768}\) is the last term.

Infinite geometric progression: An infinite geometric progression has an endless number of terms. Since there is no fixed number of terms, the last term cannot be specified. For example, the infinite series 3, -6, 12, -24, … does not have a definite end term.

GP vs AP

To help you identify the sequence effectively, we will now look at the key differences between GP and AP.

Geometric Progression (GP) Arithmetic Progression (AP) Each term is obtained by multiplying the previous term by a fixed common ratio 
𝑟. Each term is obtained by adding a fixed common difference 
𝑑 to the previous term. No common difference between the terms. There is no fixed ratio between the terms For example: 2, 4, 8, 16,...(r = 2) For example: 3, 6, 9, 12,...(d = 3) Such series can converge or diverge depending on r. The series is always divergent unless the common difference is zero. Formula for n-th term is \(a_n = a_1 \cdot r^{\,n-1} \). Formula for n-th term is \(a_n = a_1 + (n-1)d \).

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What are the Properties of GP?

Understanding the unique features of a progression helps us identify it more easily. Here are a few properties that geometric progressions (GP) follow. 

  • The square of any term in a GP is equal to the product of the terms that are directly adjacent to it: \(a_k² = a_{k-1} × a_{k+1}\)
  • In a finite geometric progression, terms that are equally spaced from the beginning and the end have the same product: \(a_1 × a_n = a_2 × a_{ n-1} =…= a_k × a_{n-k+1}\)
  • Multiplying or dividing a GP by a non-zero constant, the new sequence remains a GP with the same common ratio.
  • The reciprocal of each term in a GP results in another GP with a new common ratio equal to 1/r (r = original common ratio).
  • A GP remains a GP even if each of its terms is raised to the same power. 
    Example: GP: a, ar, ar2,...,

    Raise each term to the power k:
    \(a^k, (ar)^k, (ar²)^k, …\) which is still a GP.

What is the Formula for GP?

In a GP, the sum of the terms can be calculated using the following formulas:

For a \(GP:  a, ar, ar^2, ar^3\), …
 

  • nth term:
    \(a_n = a × r^{n-1} \space \text{or} \space a_n = r × a_{n-1}\)
     
  • Sum of the first n terms:
    \(S_n = a(1 − r^n)/(1 − r), for \space r ≠ 1\)
    \(S_n = n × a\) for r = 1
     
  • Sum of infinite terms: 
    \(S_∞ = a / (1 – r)\), when |r| < 1


The sum does not exist when \(|r| ≥ 1\).

Tips and Tricks to Master Geometric Progression

Learn how to quickly identify, analyze, and apply geometric progressions in problems and real-life scenarios.

  • Identify the common ratio to determine growth or decay.
     
  • Apply the n-th term formula.
     
  • Memorize finite and infinite sum formulas.
     
  • Determine if the series converges or diverges.
     
  • Solve real-life problems using GP concepts.

Common Mistakes and How to Avoid Them in Geometric Progression

Geometric progression is a simple mathematical concept, but many students struggle with its problems. Here are a few common mistakes and tips to avoid them:

Real-Life Applications of Geometric Progression

Geometric progression has a vital role in various real-life situations. Let's explore how this concept applies in real-life scenarios.

  • Compound Interest in Finance - The amount of money grows exponentially when interest is compounded periodically, forming a GP.
     
  • Population Growth - Populations that grow by a fixed percentage over time follow a geometric progression.
     
  • Depreciation of Assets - The value of machinery or vehicles often decreases by a fixed ratio annually, modeled using GP.
     
  • Radioactive Decay - The remaining quantity of a radioactive substance decreases by a fixed fraction over equal time intervals.
     
  • Computer Science & Algorithms - Problems like binary search or tree structures involve exponential growth or halving, which are modeled by geometric progressions.

Problem 1

Find the 5ᵗʰ term of a GP Given: First term (a) = 3 Common ratio (r) = 2

Okay, lets begin

a5 =  48

Explanation

First, apply the formula for the nᵗʰ term:
\(a_n = a \cdot r^{\,n-1} \)


Substituting the values into the formula:
\(a_5 = 3 \times 2^{\,5-1} = 3 \times 2^4 \)


Here, we get:
a5 = 3 × 16 = 48

Well explained 👍

Problem 2

Find the sum to infinity of a GP Given: a = 8, r = 1/2

Okay, lets begin

S∞ = 16

Explanation

Let’s first check if |r| < 1

It holds true for the infinite sum since |1/2| < 1.


Using the formula:
S∞ = a / (1 – r)

Substituting the values into the formula:
S∞ = 8 / (1 – 1/2) = 8 / (1/2)

So,
S∞ = 8 × 2 = 16

Well explained 👍

Problem 3

Find the sum of the first 6 terms of a GP Given: a = 5, r = 3, n = 6

Okay, lets begin

S6 = 1820

Explanation

Here, we use the formula for the sum of the first n terms

\(S_n = a(1 − r^n)/(1 − r)\)

Let’s substitute the values:
S6 = 5(36 – 1) / (3 – 1)

We now calculate powers and simplify:
36 = 729
S6 = 5 (729 – 1) / 2 = (5 × 728) / 2

So,
S6 = 3640 / 2 = 1820

Well explained 👍

Problem 4

Find the 8ᵗʰ term of the GP 5, 10, 20, 40,... Given: a = 5, r = 2, n = 8

Okay, lets begin

a8 =  640

Explanation

Here, we apply the formula for the nᵗʰ term:

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

Substituting the values into the formula:
a8 = 5 × 28 - 1 = 5 × 27

So,
a8 = 5 × 128 = 640

Well explained 👍

Problem 5

Find how many terms of the GP 3, 6, 12, 24,... are needed to make the sum 93 Given: a = 3, r = 2, Sₙ = 93

Okay, lets begin

n = 5

Explanation

Using the formula:
\(S_n = a(1 − r^n)/(1 − r)\)

Substituting the given values:
93 = 3(2n – 1) / (2 – 1)

Now, simplify to get the result:

93 = 3(2n – 1)

93 ÷ 3 = 2n – 1

31 = 2n – 1

2n = 31 + 1

2n = 32 

Here, n is the exponent to which 2 needs to be raised to obtain 32.

Since 25 = 32
→ n = 5

Well explained 👍

FAQs on Geometric Progression

1.What is meant by the term GP?

GP stands for Geometric Progression. In a GP, each term is the product of the term before it and a constant value called the common ratio.

2.Give the formula for the nth term of a GP.

The formula for the nth term is: an = a1 × r (n –1)

Here: 

an = nth term

a1 = first term

r = common ratio

n = term number

3.What do you mean by a common ratio?

The constant factor by which each term of the progression is multiplied to obtain the subsequent term is known as the common ratio (r). For example: GP: 2, 4, 8, 16; r = 2.

4.Can the common ratio be 1?

Yes, the common ratio can be 1. When r = 1, each term in the geometric progression will be the same as the first term.
Example: 5, 5, 5, 5, 5…

5.Is it possible for a geometric progression's common ratio to be negative?

Yes, a common ratio can be either positive or negative. If the common ratio is negative, then the terms in the progression will alternate between positive and negative values. 
For example: 2, -4, 8, -16, 32,…

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.