Geometric Progression Calculator – Geometric Sequence
Calculate nth term, sum, and infinite sum of geometric sequences
Table of Contents
How to Use
- Enter the first term of the sequence
- Enter the common ratio between consecutive terms
- Enter the number of terms you want to calculate
- Click calculate to see the nth term, sum, and sequence
What is a Geometric Progression?
A geometric progression (GP) or geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio.
For example: 2, 6, 18, 54, 162... is a geometric progression with first term a = 2 and common ratio r = 3.
Key Formulas
- nth term: a_n = a × r^(n-1), where a is the first term and r is the common ratio
- Sum of n terms: S_n = a × (1 - r^n) / (1 - r) when r ≠ 1
- Sum of n terms: S_n = n × a when r = 1
- Infinite sum: S_∞ = a / (1 - r) when |r| < 1
Convergence and Divergence
A geometric series converges (has a finite sum) only when the absolute value of the common ratio is less than 1 (|r| < 1). When |r| ≥ 1, the series diverges and has no finite sum.
Frequently Asked Questions
- What happens when the common ratio is 1?
- When r = 1, all terms in the sequence are equal to the first term. The sum of n terms is simply n × a, where a is the first term.
- Can a geometric progression have negative terms?
- Yes, if either the first term or common ratio (or both) is negative, the sequence will have negative terms. A negative common ratio creates an alternating sequence.
- When does the infinite sum exist?
- The infinite sum exists only when |r| < 1. In this case, the terms get smaller and smaller, approaching zero, allowing the series to converge to a finite value.