The nth term of a Geometric Progression helps you find any term in the sequence without writing all previous terms. This makes calculations faster, especially when n is large. Whether you’re solving exam problems, dealing with financial growth, or studying patterns, this formula gives a direct way to reach the required term. It is widely used in school maths, competitive exams, and real-life applications like compound interest and population models.
Formula for Calculating the nth Term of GP - Overview
| Formula | Variables | When It Is Used |
|---|---|---|
| Tn = a × r^(n – 1) | a = first term | To find the nth term in GP |
| r = common ratio | When the ratio remains constant | |
| n = term position | Used in maths, finance & exams |
What is the nth Term in GP in Maths?
The nth term of a Geometric Progression (GP) is the value of the term that appears at position n in the sequence. Instead of listing all terms one by one, we use the formula Tn = a × r^(n – 1) to find any term directly. For example, in the GP 2, 4, 8, 16…, the nth term is found using a = 2 and r = 2.
A Geometric Progression is a sequence where each term is obtained by multiplying the previous term by a constant value called the common ratio (r). The nth term formula saves time and helps in dealing with very large or very small values.
How the formula works:
- Identify the first term (a).
- Find the common ratio (r) by dividing any term by the one before it.
- Substitute values of a, r, and n into Tn = a × r^(n – 1).
Applications of nth term in GP:
- Finance: Compound interest & investments
- Science: Population growth & radioactive decay
- Exams: CUET, SSC, JEE, Banking, Class 10 & 12 maths
Examples to Calculate the nth Term of GP
Example 1: Find the 6th term of GP: 5, 15, 45, …
Step 1: a = 5, r = 3, n = 6
Step 2: Tn = 5 × 3^(6 – 1)
Step 3: = 5 × 3⁵
Step 4: = 5 × 243 = 1215
So, the 6th term is 1215.
Example 2: Find the 8th term of GP: 2, 4, 8, 16…
Step 1: a = 2, r = 2, n = 8
Step 2: Tn = 2 × 2^(8 – 1)
Step 3: = 2 × 2⁷
Step 4: = 2 × 128 = 256
So, the 8th term is 256.
FAQs about the nth Term Formula in GP
Q1. Can the nth term be negative in GP?
Yes, if the common ratio is negative, the terms alternate between positive and negative.
Q2. Does the nth term formula work for fractions?
Yes, it works even when the first term or ratio is a fraction or decimal.
Q3. What happens if r = 1?
Every term becomes equal to the first term since multiplying by 1 does not change the value.
Q4. Why is this formula important for exams?
It quickly finds any term, saving time in CUET, SSC, Banking, JEE, and school exams.
Q5. Can a GP start with zero?
Yes, but if the first term is zero, all remaining terms will also be zero.
Q6. Where is the nth term used in real life?
It appears in compound interest, physics calculations, population models, and growth/decay patterns.