A Geometric Progression (GP) is one of the most important concepts in mathematics, especially in sequences, series, and financial calculations. It represents a sequence where each term is multiplied by a constant value, called the common ratio. Understanding GP formulas makes it easy to find any term in the sequence or calculate the sum of several terms. These formulas are widely used in school maths, competitive exams, finance, compound interest, population studies, and even scientific calculations. Once you understand the logic behind GP, solving related questions becomes simple and fast.
Formula for Calculating Geometric Progression (GP) Overview
| Formula | Variables | When it is Used |
|---|---|---|
| Tn = a × r^(n – 1) | Tn = nth term | To find any term in GP |
| Sn = a × (r^n – 1) / (r – 1) (r ≠ 1) | a = first term | To calculate sum of terms |
| r = common ratio | Used in maths exams and finance | |
| n = number of terms | — |
What is GP in Maths?
A Geometric Progression (GP) is a sequence of numbers where each term is obtained by multiplying the previous term by a fixed constant, known as the common ratio (r). This ratio can be positive, negative, greater than 1, or between 0 and 1. A GP can grow very fast or decrease rapidly depending on the value of the ratio.
Example of GP:
2, 4, 8, 16… where r = 2
To find the nth term, use:
Tn = a × r^(n – 1)
To find the sum of the first n terms, use:
Sn = a × (r^n – 1) / (r – 1) (when r ≠ 1)
Steps for solving GP problems:
- Identify a (first term)
- Identify r (common ratio)
- Identify n (number of terms)
- Apply the formula correctly
- Simplify step by step
Applications of GP:
- Finance: Compound interest calculations
- Competitive exams: CUET, SSC, JEE, Banking
- Real life: Population growth, savings growth, investments, bacterial multiplication
Examples to Calculate GP
Example 1: Find the 5th term of GP – 3, 6, 12, …
Step 1: a = 3, r = 2, n = 5
Step 2: Tn = 3 × 2^(5 – 1)
= 3 × 16
= 48
So, the 5th term of the GP is 48.
Example 2: Find the sum of first 4 terms of GP – 2, 4, 8, 16
Step 1: a = 2, r = 2, n = 4
Step 2: Sn = 2 × (2^4 – 1) / (2 – 1)
= 2 × (16 – 1) / 1
= 2 × 15
= 30
So, the sum of the first 4 terms is 30.
FAQs about GP Formula
Q1. Can r be negative in GP?
Yes, if r is negative, the terms alternate between positive and negative.
Q2. What happens when r = 1?
Then every term becomes equal to the first term, so the GP becomes constant.
Q3. Is GP used in compound interest?
Yes, compound interest is directly based on GP.
Q4. How is GP different from AP?
AP has a constant difference, while GP has a constant ratio.
Q5. Does GP apply to decreasing sequences?
Yes, when 0 < r < 1, the terms of the GP keep decreasing.
Q6. Is GP important for exams?
Absolutely, GP questions frequently appear in JEE, CUET, SSC, Banking, and school-level exams.