What is a geometric sequence?

A geometric sequence is an ordered list 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. Example: 2, 6, 18, 54 … has a ratio of 3.

What formula is used for the nth term?

The nth term is calculated as a × r^(n−1), where a is the first term, r is the common ratio, and n is the position in the sequence.

When does the infinite sum converge?

The infinite sum S = a / (1 − r) only converges when the absolute value of the common ratio is less than 1 (|r| < 1). If |r| ≥ 1 the series diverges and there is no finite infinite sum.

Can the common ratio be negative or a fraction?

Yes. A negative ratio produces an alternating sequence (signs flip each term). A fractional ratio between −1 and 1 produces a decreasing sequence that converges to a finite infinite sum.

What is the partial sum formula?

The sum of the first n terms is S_n = a × (1 − r^n) / (1 − r) when r ≠ 1. When r = 1 the sum is simply a × n.

Geometric Sequence Calculator

Name: Geometric Sequence Calculator
Availability: InStock
Author: Nham Vu

Enter a first term, common ratio, and term index to find any term value and the partial or infinite series sum.

Sequence Inputs

First Term (a)

Common Ratio (r)

Term Index (n)

Must be a positive integer.

Results

Enter values and click Calculate to see results.

Formulas Used

a_n = a · rⁿ⁻¹

S_n = a · (1 − rⁿ) / (1 − r) (r ≠ 1)

S_∞ = a / (1 − r) (|r| < 1 only)

Sequence Preview (first 8 terms)

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Summary

Enter a first term, common ratio, and term index to find any term value and the partial or infinite series sum.

How it works

Enter the first term (a) of the sequence.
Enter the common ratio (r) — the multiplier between consecutive terms.
Enter the term index (n) to find — e.g., 5 for the 5th term.
Click Calculate to see the nth term, partial sum up to n, and infinite sum (when |r|<1).
Use the sequence preview to verify your inputs look correct.

Use cases

Check compound interest growth modeled as a geometric sequence.
Solve textbook problems on geometric progressions.
Calculate the total distance traveled in bouncing-ball physics problems.
Find partial sums for convergence analysis in calculus.
Model population growth or radioactive decay over discrete steps.
Verify homework answers with step-by-step formula breakdown.

Frequently Asked Questions

Last updated: 2026-06-13 · Reviewed by Nham Vu