Geometric Progression Definition Formula Properties and Solved 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.
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.
FAQs on What Are Geometric Progressions in Mathematics
1. What is a geometric progression?
A geometric progression (GP) is a sequence of numbers in which each term is obtained by multiplying the previous term by a fixed non-zero number called the common ratio.
- If the first term is a and the common ratio is r, the sequence looks like: a, ar, ar², ar³, ...
- Example: 2, 6, 18, 54 is a GP because each term is multiplied by 3.
- GP is also called a geometric sequence.
2. What is the formula for the nth term of a geometric progression?
The formula for the nth term of a geometric progression is aₙ = arⁿ⁻¹, where a is the first term and r is the common ratio.
- a = first term
- r = common ratio
- n = term number
3. How do you find the common ratio in a geometric progression?
The common ratio (r) is found by dividing any term by the previous term in a geometric progression.
- Formula: r = a₂ / a₁ or r = aₙ / aₙ₋₁
- Example: In 5, 15, 45, 135 → r = 15 ÷ 5 = 3
- The ratio must remain constant for the sequence to be a GP.
4. What is the sum of n terms of a geometric progression?
The sum of the first n terms of a geometric progression is given by Sₙ = a(1 − rⁿ) / (1 − r) when r ≠ 1.
- a = first term
- r = common ratio
- n = number of terms
5. What is the formula for the sum to infinity of a geometric progression?
The sum to infinity of a geometric progression is S∞ = a / (1 − r) when |r| < 1.
- This formula works only if the common ratio satisfies |r| < 1.
- Example: If a = 4 and r = 1/2, then S∞ = 4 / (1 − 1/2) = 8.
- If |r| ≥ 1, the sum to infinity does not exist.
6. What is the difference between arithmetic progression and geometric progression?
The main difference is that an arithmetic progression (AP) adds a constant difference, while a geometric progression (GP) multiplies by a constant ratio.
- AP rule: add a fixed number (common difference).
- GP rule: multiply by a fixed number (common ratio).
- Example AP: 2, 5, 8, 11 (add 3).
- Example GP: 2, 6, 18, 54 (multiply by 3).
7. How do you check if a sequence is a geometric progression?
A sequence is a geometric progression if the ratio of consecutive terms is constant.
- Step 1: Divide the second term by the first.
- Step 2: Divide the third term by the second.
- Step 3: If all ratios are equal, it is a GP.
8. Can a geometric progression have a negative common ratio?
Yes, a geometric progression can have a negative common ratio, which causes the terms to alternate in sign.
- Example: 2, −4, 8, −16
- Here, r = −4 ÷ 2 = −2
- The signs alternate because each term is multiplied by −2.
9. What are some real-life applications of geometric progression?
Geometric progression is used to model situations involving repeated multiplication or exponential growth and decay.
- Compound interest in finance
- Population growth models
- Radioactive decay
- Depreciation of assets
10. What are the important properties of a geometric progression?
The key properties of a geometric progression involve its constant ratio and term relationships.
- Each term = previous term × r
- The nth term formula is aₙ = arⁿ⁻¹
- If three numbers a, b, c are in GP, then b² = ac
- The sum to infinity exists only if |r| < 1
