What Is a Geometric Progression? (With Examples)

The concept of geometric progression is essential in mathematics and helps in solving real-world and exam-level problems efficiently.

Understanding Geometric Progression

A geometric progression (often called GP) refers to a sequence of numbers in which each term after the first is found by multiplying the previous one by a constant called the common ratio. This concept is widely used in finance (such as interest calculations), science (like population growth), and statistics (for modeling exponential changes).

Definition and Key Identifiers of Geometric Progression

A geometric progression can be identified by the following properties:

• Each term (except the first) is the previous term multiplied by a fixed constant (the common ratio, r).
• The pattern continues throughout the sequence.
• The sequence can be finite (ends after a certain number of terms) or infinite.
• The common ratio can be positive or negative, but is never zero.

For example, 2, 4, 8, 16 is a geometric progression with a common ratio of 2.

Geometric Progression vs Arithmetic Progression

Students often confuse arithmetic progression (AP) with geometric progression (GP). Here’s a quick comparison:

Feature	Geometric Progression (GP)	Arithmetic Progression (AP)
How is the next term found?	Multiply by constant ratio	Add or subtract constant difference
General form	a, ar, ar2, ar3, ...	a, a+d, a+2d, ...
Common Ratio / Difference	Common ratio (r)	Common difference (d)
Growth graph	Exponential	Linear

For a deeper comparison, visit Arithmetic Progression.

Formula Used in Geometric Progression

The standard formulas for a geometric progression are:

• General Term (nth term): \( a_n = a \cdot r^{n-1} \)
  
  Where 'a' = first term and 'r' = common ratio.
• Sum of first n terms (finite GP): \( S_n = a \frac{r^n - 1}{r - 1} \) (for r ≠ 1)
• Sum of infinite GP: \( S_\infty = \frac{a}{1 - r} \), only if |r| < 1.

These formulas form the foundation for solving all types of geometric progression questions.

Here’s a Helpful Example Table

Sequence	Common Ratio	Is it GP?
3, 6, 12, 24	2	Yes
5, 10, 15, 20	2, then 1.5, ...	No
81, 27, 9, 3	1/3	Yes

This table shows how a geometric progression always follows a fixed multiplying pattern.

Worked Example – Solving a Problem

Let’s determine if the sequence 2, 8, 32, 128 is a geometric progression:

1. Find the ratio of the second term to the first: 8 ÷ 2 = 4.

2. Find the ratio of the third term to the second: 32 ÷ 8 = 4.

3. Find the ratio of the fourth term to the third: 128 ÷ 32 = 4.

4. Since the common ratio is always 4, this is a geometric progression.

Answer: Yes, 2, 8, 32, 128 is a geometric progression with common ratio 4.

Another Example – Find the 5th Term

Given: First term a = 2, common ratio r = 3.
Find the 5th term.

1. Use the formula: \( a_n = a \cdot r^{n-1} \)

2. Substitute the values: \( a_5 = 2 \cdot 3^{4} \)

3. Calculate powers: \( 3^{4} = 81 \)

4. Multiply: \( 2 \cdot 81 = 162 \)

Answer: The 5th term is 162.

Practice Problems

• Find the 7th term of the GP: 5, 10, 20, ...
• Determine if the sequence 1, 2, 4, 8, 15 is a geometric progression.
• Calculate the sum of first 6 terms of GP with a = 3, r = 2.
• List any three real-life applications of geometric progression.

You can get more questions and answers for practice from our Sequences and Series Practice Paper.

Common Mistakes to Avoid

• Confusing geometric progression with arithmetic progression (using addition instead of multiplication).
• Forgetting to check if all ratios are the same throughout the sequence.
• Using the infinite sum formula when |r| ≥ 1, which is not allowed.

Real-World Applications

The concept of geometric progression appears in areas such as:

• Compound interest calculations in banking
• Population growth modelling
• Radioactive decay in chemistry/physics
• Growth or decay of investments and depreciation
• Repeated doubling or halving in computer algorithms

Vedantu helps students see how these maths patterns are used beyond the classroom, preparing them for board exams and competitive tests. For more on practical uses, check Applications of Mathematics.

Quick Revision Table

Item	GP Formula / Point
nth Term	\( a_n = a \cdot r^{n-1} \)
Sum of First n Terms	\( S_n = a \frac{r^n - 1}{r-1} \) (r ≠ 1)
Sum to Infinity	\( S_\infty = \frac{a}{1-r} \), |r| < 1
Common Ratio	Divide any term by its previous one
GP Form Pattern	a, ar, ar2, ar3, ...

Key Internal Resources for Further Study

• Sum of n Terms
• Sequences and Series
• Geometric Progression Sum of GP
• Arithmetic Geometric Sequence
• Binomial Theorem
• Arithmetic Progression
• Applications of Mathematics

We explored the idea of geometric progression, how to apply it, solve related problems, and understand its real-life relevance. Practice more with Vedantu to build confidence in these concepts.