# What is my arithmetic sequence?

Finds the nth term and the sum of the first n terms of an arithmetic sequence from its first term and common difference, with the first terms and the explicit and recursive formulas.

- Page: https://www.acalculator.org/math/arithmetic-sequence-calculator
- JSON spec: https://www.acalculator.org/math/arithmetic-sequence-calculator.json
- Version: 61d63c1cf6c0

## Default answer

Example with the default inputs (First term (a₁) 2, Common difference (d) 3, Term number (n) 10): Term 10 is 29, and the sum of terms 1 to 10 is 155.

## Inputs

| Key | Label | Description |
| --- | --- | --- |
| a | First term (a₁) | The first term of the sequence. |
| d | Common difference (d) | What is added to each term 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. |

## Method

aₙ = a₁ + (n − 1)d; Sₙ = n(a₁ + aₙ)/2.

## 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 = 2, d = 10, n = 5 gives nth = 42, sum = 110, terms = 2, 12, 22, 32, 42, explicit = a_n = 2 + 10(n - 1). Source: OpenStax, Algebra and Trigonometry 2e, section 13.2 Arithmetic Sequences (aₙ = a₁ + d(n − 1)), https://openstax.org/books/algebra-and-trigonometry-2e/pages/13-2-arithmetic-sequences, retrieved 2026-10-02, example: 2, 12, 22, 32, 42 has aₙ = 2 + 10(n − 1) and a₅ = 42.
2. a = 5, d = 3, n = 10 gives nth = 32, sum = 185, recursive = a_1 = 5, a_n = a_(n - 1) + 3. Source: OpenStax, Algebra and Trigonometry 2e, section 13.4 Series and Their Notations (Sₙ = n(a₁ + aₙ)/2), https://openstax.org/books/algebra-and-trigonometry-2e/pages/13-4-series-and-their-notations, retrieved 2026-10-02, example: 5 + 8 + … + 32 = 185.
3. a = 8, d = 2, n = 5 gives nth = 16, sum = 60. Source: OpenStax, Algebra and Trigonometry 2e, section 13.2 Arithmetic Sequences (aₙ = a₁ + d(n − 1)), https://openstax.org/books/algebra-and-trigonometry-2e/pages/13-2-arithmetic-sequences, retrieved 2026-10-02, example: a₁ = 8, d = 2 gives a₅ = 16.
4. a = 10, d = -4, n = 6 gives nth = -10, sum = 0, explicit = a_n = 10 - 4(n - 1), terms = 10, 6, 2, -2, -6, -10.
5. a = 0.1, d = 0.2, n = 3 gives nth = 0.5, sum = 0.9, terms = 0.1, 0.3, 0.5.

## FAQ

### What is an arithmetic sequence?

A list of numbers where the same amount, the common difference d, is added each time. 2, 12, 22, 32, 42 adds 10 each time, so d = 10. A negative d makes the terms fall: 10, 6, 2, −2.

### How do I find the nth term?

Use aₙ = a₁ + (n − 1)d. For 2, 12, 22, …, the 5th term is 2 + 4 × 10 = 42. The formula simplifies to aₙ = 10n − 8.

### How do I find the common difference?

Subtract any term from the next one: 12 − 2 = 10. If you know two terms that are not next to each other, divide their difference by the gap in positions: with a₁ = 8 and a₄ = 14, d = (14 − 8) ÷ 3 = 2.

### How do I add up an arithmetic sequence?

Multiply the number of terms by the average of the first and last terms: Sₙ = n(a₁ + aₙ)/2. For 5 + 8 + 11 + … + 32 (10 terms): 10 × (5 + 32) ÷ 2 = 185.

### What is the difference between an arithmetic sequence and an arithmetic series?

The sequence is the list of terms (5, 8, 11, …); the series is their sum (5 + 8 + 11 + …). The calculator gives both: the terms and the sum of the first n.

### What is the recursive formula?

The first term and the rule for each next term: a₁ = 5, aₙ = aₙ₋₁ + 3. The explicit formula, aₙ = 5 + 3(n − 1), gives any term straight from n.

## Sources

- OpenStax, Algebra and Trigonometry 2e, section 13.2 Arithmetic Sequences (aₙ = a₁ + d(n − 1); examples 2, 12, 22, … and a₁ = 8, a₄ = 14), CC BY 4.0, retrieved 2026-10-02. https://openstax.org/books/algebra-and-trigonometry-2e/pages/13-2-arithmetic-sequences
- OpenStax, Algebra and Trigonometry 2e, section 13.4 Series and Their Notations (Sₙ = n(a₁ + aₙ)/2; example 5 + 8 + … + 32 = 185), CC BY 4.0, retrieved 2026-10-02. https://openstax.org/books/algebra-and-trigonometry-2e/pages/13-4-series-and-their-notations