# Sequence Calculator

URL: https://speedycalculator.com/calculators/math/sequence/
Last updated: 2026-10-30

> The sequence calculator lists terms of ten common sequences – arithmetic, geometric, Fibonacci, primes, powers of 2, factorials, triangular, square, cube and star numbers – starting from any index.

## Inputs

- Sequence
- First term a₀
- Difference d / ratio r
- Start at index n
- Number of terms

## Results

- Terms
- Rule
- Term at the starting index

## Formula

```
Arithmetic aₙ = a₀ + nd · Geometric aₙ = a₀rⁿ · Triangular n(n + 1)/2 · Star 6n(n − 1) + 1
```

## Explanation

Sequences are ordered lists of numbers following a rule. Recognising which rule a list follows is a common puzzle and exam question, and many sequences – primes, triangular numbers, factorials – have their own entries in the On-Line Encyclopedia of Integer Sequences.

### How to use the Sequence Calculator

- **Sequence**
- **First term a₀**
- **Difference d / ratio r**
- **Start at index n**
- **Number of terms**

Everything is calculated in your browser, so results appear instantly and nothing you type is sent to a server.

### How it works

Choose the sequence, the starting index n and how many terms to show. For arithmetic and geometric sequences you also give the first term and the difference or ratio.

### Tips

- Write out the first few terms to check the pattern before using a formula.
- Decide whether the sequence starts at term 0 or term 1 – formulas differ by one place.
- Arithmetic sequences add a constant; geometric sequences multiply by one.
- An infinite geometric series only has a sum when |r| < 1.

### Common mistakes to avoid

- Off-by-one errors in the term number.
- Using the arithmetic sum formula for a geometric sequence.
- Ignoring the sign of a negative ratio or difference.

## Worked example

Inputs — Sequence: Arithmetic; First term a₀: 2; Difference d / ratio r: 3; Start at index n: 0; Number of terms: 10.

Results — **Terms = 2, 5, 8, 11, 14, 17, 20, 23, 26, 29**; Rule = aₙ = a₀ + n·d; Term at the starting index = 2.

## Key terms

- **Sequence**: An ordered list of numbers.

## FAQ

**What is a star number?**
A centred hexagram number: 1, 13, 37, 73, …

**Are prime numbers a sequence?**
Yes, though there is no simple formula for the n-th prime.

**What is the difference between a sequence and a series?**
A series is the sum of a sequence's terms.

**Why start at index 0?**
Many sequences (like Fibonacci) are conventionally indexed from 0.

**What formula does the Sequence Calculator use?**
Arithmetic aₙ = a₀ + nd · Geometric aₙ = a₀rⁿ · Triangular n(n + 1)/2 · Star 6n(n − 1) + 1.


## Sources

- [OpenStax – College Algebra 2e](https://openstax.org/details/books/college-algebra-2e)
- [OpenStax – Calculus Volume 2](https://openstax.org/details/books/calculus-volume-2)
- [OEIS – The On-Line Encyclopedia of Integer Sequences](https://oeis.org/)