Q1. Find the sum of the GP 2, 6, 18, … up to 5 terms.
a = 2, r = 3, n = 5
Sₙ = a(rⁿ − 1)/(r − 1) = 2(3⁵ − 1)/(3 − 1)
S = 2(243 − 1)/2 = 242
Sequences and Series — Class 11
From AP to GP, every sequence has an nth-term and a sum formula. Compute all four in one place.
Arithmetic Progression (AP)Constant difference d: aₙ = a + (n−1)d, Sₙ = n/2[2a + (n−1)d].
Geometric Progression (GP)Constant ratio r: aₙ = a·rⁿ⁻¹, Sₙ = a(rⁿ − 1)/(r − 1) for r ≠ 1.
MeansArithmetic mean of a, b = (a+b)/2; geometric mean = √(ab).
AP & GP calculator Interactive
Enter the first term, common difference d (AP), common ratio r (GP) and n.
AP nth term—
AP sum—
GP nth term—
GP sum—
An infinite GP with |r| < 1 has a finite sum S∞ = a/(1 − r) — used in recurring-decimal problems.
Solved examples
Formula sheet
| Quantity | Formula |
|---|---|
| AP nth term | aₙ = a + (n−1)d |
| AP sum | Sₙ = n/2 [2a + (n−1)d] |
| GP nth term | aₙ = a·rⁿ⁻¹ |
| GP sum | Sₙ = a(rⁿ − 1)/(r − 1) |
| Sum to infinity (GP) | S∞ = a/(1 − r), |r| < 1 |
Common mistakes & exam wins
- Check whether it is an AP (constant difference) or a GP (constant ratio) first.
- For a GP with r = 1, the sum is simply n·a.
- Sum of first n natural numbers = n(n+1)/2; of squares = n(n+1)(2n+1)/6.
Frequently asked questions
What is the sum of n terms of a GP?
Sₙ = a(rⁿ − 1)/(r − 1) for r ≠ 1, where a is the first term and r the common ratio.
What is the sum of an infinite GP?
S∞ = a/(1 − r), valid only when |r| < 1.
What is the geometric mean of two numbers?
The geometric mean of a and b is √(ab).