The formula for calculating Geometric Progression (GP) helps us find terms or the sum of terms in a sequence where each number is multiplied by a fixed value.
The nth term of a GP is:
Tn = a × r^(n – 1)
Where:
- Tn = nth term
- a = first term
- r = common ratio
- n = number of terms
For the sum of n terms (when r ≠ 1):
Sn = a × (r^n – 1) / (r – 1)
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) | 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?
In mathematics, a Geometric Progression (GP) is a sequence where each term is obtained by multiplying the previous term by a fixed number (ratio).
Example: 2, 4, 8, 16… (common ratio r = 2).
A GP grows or decreases rapidly depending on the ratio.
- To find the nth term: use Tn = a × r^(n – 1).
- To find the sum: use Sn = a × (r^n – 1) / (r – 1) when r ≠ 1.
Steps:
- Identify a (first term), r (common ratio), and n (number of terms).
- Apply the formula step by step.
- Simplify carefully.
Applications:
- Finance: Compound interest follows GP.
- Exams: Questions in CUET, SSC, Banking, JEE.
- Real life: Population growth, investments, bacterial growth, etc.
Examples for calculating nth term and sum of a GP
Example 1 (nth term):
Find the 5th term of GP: 3, 6, 12, …
Solution :
Step 1: a = 3, r = 2, n = 5
Step 2: Tn = 3 × 2^(5 – 1)
= 3 × 16
= 48
So, the 5th term of GP: 3, 6, 12, … is 48
Example 2 (sum):
Find the sum of first 4 terms of GP: 2, 4, 8, 16.
Solution :
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 first 4 terms of GP: 2, 4, 8, 16. is 30
FAQs about GP
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?
The GP becomes constant, and every term equals the first term a.
Q3. Is GP used in compound interest?
Yes, compound interest is a real-life application of GP where money grows by a fixed rate.
Q4. How is GP different from AP?
In AP, the difference is constant, while in GP, the ratio is constant.
Q5. Does GP apply to decreasing sequences?
Yes, when 0 < r < 1, the GP decreases with each term.
Q6. Is GP important for exams?
Yes, GP problems are frequent in CUET, SSC, Banking, JEE, and other competitive exams.