# Geometric Sequence Calculator

> nth term, partial sum and infinite sum of a geometric sequence from the first term and ratio.

Page: https://freecalculator.tech/tools/geometric-sequence-calculator
Last reviewed: 2026-11-03

## Formula

aₙ = a₁ × r^(n−1); Sₙ = a₁(1 − rⁿ) ÷ (1 − r) for r ≠ 1 (Sₙ = n a₁ when r = 1); infinite sum = a₁ ÷ (1 − r) when |r| < 1.

- Source: OpenStax College Algebra 2e §9.3 Geometric Sequences and §9.4 Series.
- aₙ = a₁ × r^(n−1); Sₙ = a₁(1 − rⁿ) ÷ (1 − r) for r ≠ 1 (Sₙ = n a₁ when r = 1); infinite sum = a₁ ÷ (1 − r) when |r| < 1.
- Inputs used: First term a₁, Common ratio r, Term number n.

## Inputs

- First term a₁
- Common ratio r
- Term number n

## Worked example: 2, 6, 18, … to the 5th term

a₅ = 162 and S₅ = 242.
1. a₅ = 2 × 3⁴ = 162.
2. S₅ = 2(1 − 243) ÷ (1 − 3) = 242.

## Common questions

### When does an infinite geometric series converge?

Only when |r| < 1; the sum is a₁ ÷ (1 − r).

### Can r be negative?

Yes. Terms then alternate in sign.

### How do I find r?

Divide any term by the previous one.

### How do I use the Geometric Sequence Calculator?

Enter the required values for First term a₁, Common ratio r, Term number n. The calculator applies the formula on this page and shows the main result with any supporting breakdown so you can check the arithmetic.

### What formula does the Geometric Sequence Calculator use?

aₙ = a₁ × r^(n−1); Sₙ = a₁(1 − rⁿ) ÷ (1 − r) for r ≠ 1 (Sₙ = n a₁ when r = 1); infinite sum = a₁ ÷ (1 − r) when |r| < 1. The visible formula section above lists the calculation path and the edge cases the page handles, so the result can be checked without relying on the form alone.

### Can the Geometric Sequence Calculator be used for exact decisions?

Use it as a calculation aid, not as a substitute for checking the underlying rule, contract, policy, or professional advice that applies to your situation. When a result depends on local rules, personal details, prices, or dates, enter those values yourself and confirm them before acting.

## Limitations and assumptions

- Very large n or r can overflow; the calculator then reports an out-of-range result.
- Floating-point rounding applies.
- Exact mathematical relationships, rounded for display. Check units and rounding before using a result in graded, engineering or financial work.
- The result depends on the values you enter for this geometric sequence calculator; it does not supply missing rates, rules, prices, dates, or assumptions for you.

## Sources

- OpenStax — https://openstax.org/books/college-algebra-2e/pages/9-3-geometric-sequences

## Cite this page

- APA: Free Calculator. (2026, November 3). Geometric Sequence Calculator. https://freecalculator.tech/tools/geometric-sequence-calculator
- MLA: "Geometric Sequence Calculator." Free Calculator, 3 Nov. 2026, https://freecalculator.tech/tools/geometric-sequence-calculator.
- Link: Geometric Sequence Calculator — Free Calculator, https://freecalculator.tech/tools/geometric-sequence-calculator