The sum of n terms of a Geometric Progression (GP) helps you calculate the total of the first few terms of a sequence where each term is multiplied by a fixed ratio. Instead of adding terms manually, this formula gives a quick and accurate shortcut. It is widely used in maths chapters on sequences and series, as well as real-life calculations involving growth, interest, and repeated multiplication patterns. This concept also appears frequently in exams like CUET, SSC, Banking, and JEE.
Formula for Calculating Sum of n Terms of a GP – Overview
| Formula | Variables | When It Is Used |
|---|---|---|
| Sn = a × (rⁿ – 1) / (r – 1) | a = first term, r = common ratio, n = number of terms | Used to find the total of the first n terms of a GP |
What is Sum of n Terms of a GP in Maths?
The sum of n terms of a GP refers to the total you get when you add the first n numbers of a geometric sequence. In this sequence, each term is obtained by multiplying the previous term by a constant value called the common ratio (r).
To calculate the sum, the formula Sn = a × (rⁿ – 1) / (r – 1) is used when r ≠ 1, because the series grows or shrinks based on multiplication. If r = 1, the terms remain the same throughout, so the sum becomes Sn = a × n.
This formula is derived by writing the series twice, multiplying, and subtracting equations to eliminate repetitive terms. It is used in calculating growth, compound interest, population patterns, and many competitive exam problems.
Examples to Calculate Sum of n Terms of a GP
Example 1: GP = 3, 6, 12, 24… (n = 5)
Step 1: a = 3, r = 2, n = 5
Step 2: Sn = a × (rⁿ – 1) / (r – 1)
= 3 × (2⁵ – 1) / (2 – 1)
= 3 × (32 – 1) / 1
= 3 × 31
Result: 93
So, the sum of the first 5 terms is 93.
Example 2: GP = 5, 15, 45, 135… (n = 4)
Step 1: a = 5, r = 3, n = 4
Step 2: Sn = 5 × (3⁴ – 1) / (3 – 1)
= 5 × (81 – 1) / 2
= 5 × 80 / 2
= 5 × 40
Result: 200
So, the sum of the first 4 terms is 200.
FAQs about Sum of n Terms of a GP Formula
Q1. What if the common ratio r = 1 in GP?
If r = 1, all terms are the same, so the sum becomes Sn = a × n.
Q2. Where is the GP sum formula used in real life?
In compound interest, population growth, depreciation, financial modelling, and repeated growth problems.
Q3. Is GP important for competitive exams?
Yes, it frequently appears in CUET, SSC, Banking, JEE, and school-level exams.
Q4. How is the AP sum formula different from GP?
AP uses constant addition (difference), while GP uses constant multiplication (ratio). Hence the formulas differ.
Q5. Can GP include fractions or decimals?
Yes, GP can have fractional, decimal, or even negative terms as long as the ratio remains constant.