# Geometric Sequence Calculator

Calculate the nth term, partial sum, and infinite sum of a geometric sequence. Enter the first term and common ratio.

## What this calculates

Find any term, the partial sum, and the infinite sum of a geometric sequence. Enter the first term, common ratio, and the desired term number to see the results and a preview of the sequence.

## Inputs

- **First Term (a₁)** — The first term of the geometric sequence.
- **Common Ratio (r)** — The constant ratio between consecutive terms.
- **Term Number (n)** — min 1 — Which term to calculate (n ≥ 1).

## Outputs

- **nth Term (aₙ)** — The value of the nth term: a₁ × r^(n-1).
- **Sum of First n Terms (Sₙ)** — The sum of all terms from a₁ to aₙ.
- **Infinite Sum (S∞)** — formatted as text — The sum of the infinite series (only converges when |r| < 1).
- **Sequence Preview** — formatted as text — The first 5 terms of the sequence.

## Details

A geometric sequence (or geometric progression) is a sequence where each term is found by multiplying the previous term by a fixed constant called the common ratio (r). For example, the sequence 3, 6, 12, 24, 48 has a common ratio of 2.

The nth term is aₙ = a₁ × r^(n-1). The sum of the first n terms is Sₙ = a₁(1 - r^n) / (1 - r) when r is not equal to 1. If |r| < 1, the infinite geometric series converges to S∞ = a₁ / (1 - r). For example, the series 1 + 1/2 + 1/4 + 1/8 + ... converges to 2.

Geometric sequences model exponential growth and decay, including compound interest, population growth, radioactive decay, and the physics of bouncing balls. They are also fundamental to the study of infinite series, fractals, and Zeno's paradox.

## Frequently Asked Questions

**Q: What is a geometric sequence?**

A: A geometric sequence is a list of numbers where each term is obtained by multiplying the previous term by a fixed value called the common ratio (r). Examples: 2, 6, 18, 54 (r = 3) and 100, 50, 25, 12.5 (r = 0.5).

**Q: When does the infinite geometric series converge?**

A: The infinite sum converges (has a finite value) only when the absolute value of the common ratio is less than 1 (|r| < 1). The sum is S∞ = a₁ / (1 - r). When |r| ≥ 1, the series diverges to infinity.

**Q: How do I find the common ratio?**

A: Divide any term by its preceding term: r = aₙ / aₙ₋₁. For the sequence 5, 15, 45, 135, the common ratio is 15/5 = 3. All consecutive term ratios should give the same value in a true geometric sequence.

**Q: What is the difference between arithmetic and geometric sequences?**

A: An arithmetic sequence adds a constant difference between terms (linear growth). A geometric sequence multiplies by a constant ratio (exponential growth or decay). Arithmetic: 2, 5, 8, 11. Geometric: 2, 6, 18, 54.

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Source: https://vastcalc.com/calculators/math/geometric-sequence
Category: Math
Last updated: 2026-04-21