Geometric sequence calculator
In a geometric sequence each term is multiplied by a fixed ratio, like 2, 6, 18. Enter the first term, the ratio and n to get the term and the sum.
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How to use it
- Enter first term a₁, common ratio r and number of terms n.
- The result appears straight away and updates as you edit.
- Copy the result, or copy a link that keeps the same values to share it.
How it is calculated
aₙ = a₁ × r^(n - 1). Sum: Sₙ = a₁ × (1 - rⁿ) ÷ (1 - r) for r ≠ 1, and Sₙ = n × a₁ for r = 1. Infinite sum = a₁ ÷ (1 - r) when |r| < 1.
Example
Inputs: First term a₁ = 2, Common ratio r = 3, Number of terms n = 5. Result: Term 5: 162; Sum of the first 5 terms: 242; Infinite sum: Does not converge (|r| ≥ 1).
Limits and caveats
- Values use standard floating point precision (about 15 significant digits).
- An infinite sum exists only when the absolute value of the ratio is below 1.
Questions about this tool
How do I find the common ratio?
Divide any term by the one before it. In 2, 6, 18 the ratio is 6 ÷ 2 = 3.
When does the series have an infinite sum?
When |r| is below 1 the terms shrink and the sum approaches a₁ ÷ (1 - r). For example 1 + 0.5 + 0.25 approaches 2.
How is it different from an arithmetic sequence?
An arithmetic sequence adds a fixed amount each step; a geometric one multiplies by a fixed ratio.