Overview
A geometric sequence is a list of numbers where each term is multiplied by the common ratio to get the next term. When the ratio lies between −1 and 1, the terms approach zero and a finite infinite sum can be computed.
Formula
aₙ = a₁ · r^(n−1). Sₙ = a₁(rⁿ − 1) ÷ (r − 1) for r ≠ 1, and S∞ = a₁ ÷ (1 − r) when |r| < 1.
How it works
Identify the inputs
The first term a₁, ratio r and number of terms n.
Compute the nth term
Multiply the first term by r raised to n − 1.
Compute the sum
Use the appropriate sum formula, with a special case when r = 1.
Check convergence
If |r| < 1, the infinite sum can be computed.
Worked examples
Example 1: a₁ = 2, r = 3, n = 5
a₅ = 2×3⁴ = 162, and the sum = 2(3⁵ − 1) ÷ 2 = 242.
Example 2: a₁ = 1, r = 0.5
The infinite sum = 1 ÷ (1 − 0.5) = 2.
Frequently asked questions
What does convergence mean?
The sum of the terms approaches a fixed value as the number of terms grows.
What happens when r = 1?
All terms are equal, the sum is n × a₁, and the usual ratio formula does not apply.
Can the ratio be negative?
Yes, and the signs of the terms alternate between positive and negative.
This calculator is for educational and informational purposes only and is not a substitute for professional advice.