# What is my geometric sequence?

Finds the nth term, the sum of the first n terms, and the sum to infinity of a geometric sequence from its first term and common ratio, with the first terms and the formulas.

- Page: https://www.acalculator.org/math/geometric-sequence-calculator
- JSON spec: https://www.acalculator.org/math/geometric-sequence-calculator.json
- Version: c9a876bbee80

## Default answer

Example with the default inputs (First term (a₁) 2, Common ratio (r) 2, Term number (n) 10): Term 10 is 1,024, and the sum of terms 1 to 10 is 2,046.

## Inputs

| Key | Label | Description |
| --- | --- | --- |
| a | First term (a₁) | The first term of the sequence. |
| r | Common ratio (r) | What each term is multiplied by to get the next one. |
| n | Term number (n) | Which term to find, and how many terms to add up, from 1 to 10,000. |

## Outputs

| Key | Label | Description |
| --- | --- | --- |
| nth | nth term (aₙ) | The value of term number n. |
| sum | Sum of the first n terms (Sₙ) | a₁ + a₂ + … + aₙ. |
| terms | First terms | The first terms of the sequence, up to 10 of them. |
| explicit | Explicit formula | A formula for term number n on its own. |
| recursive | Recursive formula | The first term, and how each term follows from the one before. |
| infinite | Sum to infinity | For a geometric sequence with −1 < r < 1, the limit of the sums: a₁ ÷ (1 − r). |

## Method

aₙ = a₁ × r^(n − 1); Sₙ = a₁(1 − rⁿ)/(1 − r), or n × a₁ when r = 1; for −1 < r < 1 the sum to infinity is a₁/(1 − r).

## Assumptions

- Terms are numbered from 1; the first term is a₁.
- The numbers are read as the exact decimals typed and the formulas are worked out in exact fractions; each result is then the nearest 64-bit float.
- A term or sum beyond about 1.8 × 10^308 has no answer.

## Worked examples

1. a = 8, r = -0.5, n = 11 gives nth = 0.007813, sum = 5.335938, infinite = 5.333333, recursive = a_1 = 8, a_n = (-0.5) × a_(n - 1). Source: OpenStax, Algebra and Trigonometry 2e, section 13.4 Series and Their Notations (Sₙ = a₁(1 − rⁿ)/(1 − r), S = a₁/(1 − r) for |r| < 1), https://openstax.org/books/algebra-and-trigonometry-2e/pages/13-4-series-and-their-notations, retrieved 2026-10-02, example: 8 + (−4) + 2 + … has S₁₁ ≈ 5.336.
2. a = 248.6, r = 0.4, n = 3 gives nth = 39.776, sum = 387.816, infinite = 414.333333, terms = 248.6, 99.44, 39.776. Source: OpenStax, Algebra and Trigonometry 2e, section 13.4 Series and Their Notations (Sₙ = a₁(1 − rⁿ)/(1 − r), S = a₁/(1 − r) for |r| < 1), https://openstax.org/books/algebra-and-trigonometry-2e/pages/13-4-series-and-their-notations, retrieved 2026-10-02, example: 248.6 + 99.44 + 39.776 + … = 414.3̅.
3. a = 3, r = 2, n = 8 gives nth = 384, sum = 765, terms = 3, 6, 12, 24, 48, 96, 192, 384, explicit = a_n = 3 × 2^(n - 1). Source: OpenStax, Algebra and Trigonometry 2e, section 13.3 Geometric Sequences (aₙ = a₁rⁿ⁻¹), https://openstax.org/books/algebra-and-trigonometry-2e/pages/13-3-geometric-sequences, retrieved 2026-10-02.
4. a = 5, r = -3, n = 4 gives nth = -135, sum = -100.

## FAQ

### What is a geometric sequence?

A list of numbers where each term is the one before times the same number, the common ratio r. 3, 6, 12, 24 has r = 2; 8, −4, 2, −1 has r = −1/2, so its signs alternate.

### How do I find the nth term?

Use aₙ = a₁ × r^(n − 1). For 3, 6, 12, …, the 8th term is 3 × 2⁷ = 384.

### How do I find the common ratio?

Divide any term by the one before it: 6 ÷ 3 = 2. Check a second pair to be sure the ratio stays the same; if it does not, the sequence is not geometric.

### How do I add up the first n terms?

Use Sₙ = a₁(1 − rⁿ)/(1 − r) when r is not 1. For 8 + (−4) + 2 + … (11 terms), S₁₁ = 8(1 − (−1/2)¹¹)/(1 + 1/2) = 5.3359375. When r = 1 every term is a₁, so Sₙ = n × a₁.

### When does a geometric series have a sum to infinity?

Only when −1 < r < 1. The terms then shrink toward 0, and the sums approach a₁/(1 − r). For 248.6 + 99.44 + 39.776 + … (r = 0.4), the sum to infinity is 248.6 ÷ 0.6 = 414.33…. With r = 2 or r = −1 the sums never settle, so there is none.

### Why does the sum to infinity matter?

It turns an endless repeating pattern into one number. A repeating decimal is an example: 0.333… is 0.3 + 0.03 + 0.003 + …, a geometric series with r = 0.1, whose sum is 0.3 ÷ 0.9 = 1/3.

## Sources

- OpenStax, Algebra and Trigonometry 2e, section 13.3 Geometric Sequences (aₙ = a₁rⁿ⁻¹), CC BY 4.0, retrieved 2026-10-02. https://openstax.org/books/algebra-and-trigonometry-2e/pages/13-3-geometric-sequences
- OpenStax, Algebra and Trigonometry 2e, section 13.4 Series and Their Notations (Sₙ = a₁(1 − rⁿ)/(1 − r); S = a₁/(1 − r) for −1 < r < 1; examples 8 + (−4) + 2 + … and 248.6 + 99.44 + 39.776 + …), CC BY 4.0, retrieved 2026-10-02. https://openstax.org/books/algebra-and-trigonometry-2e/pages/13-4-series-and-their-notations