# Geometric Sequence Calculator

URL: https://speedycalculator.com/calculators/math/geometric-sequence/
Last updated: 2026-10-25

> The geometric sequence calculator finds the nth term, the sum of the first n terms and, when it exists, the sum to infinity of a sequence with a constant ratio.

## Inputs

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

## Results

- nth term
- Sum of first n terms
- Sum to infinity

## Formula

```
aₙ = a₁ × r^(n − 1)
Sₙ = a₁(1 − rⁿ) ÷ (1 − r)
S∞ = a₁ ÷ (1 − r), |r| < 1
```

## Explanation

Geometric sequences multiply by the same ratio each step, like 2, 6, 18, 54, …

### How to use the Geometric Sequence Calculator

- **First term (a₁)**
- **Common ratio (r)**
- **Term number (n)**

Results update as you type, and every field accepts the units shown in its menu.

### How it works

Raise the ratio to n − 1 and multiply by the first term. The sum uses the closed formula for a geometric series.

### Inputs explained

FieldWhat to enterFirst term (a₁)A numberCommon ratio (r)A numberTerm number (n)A number

### Example values

How the main result changes when **First term** changes and everything else stays at the example values above:

First term (a₁)nth term12,1871.53,280.524,3742.55,467.536,56148,748

## Worked example

Inputs — First term (a₁): 2; Common ratio (r): 3; Term number (n): 8.

Results — **nth term = 4,374**; Sum of first n terms = 6,560.

## Key terms

- **Common ratio**: The fixed multiplier between consecutive terms.

## FAQ

**When does the infinite sum exist?**
Only when the ratio is between −1 and 1.

**Where are geometric sequences used?**
Compound interest, population growth and fractals.

**What formula does the Geometric Sequence Calculator use?**
aₙ = a₁ × r^(n − 1).