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.
Formula
Arithmetic aₙ = a₀ + nd · Geometric aₙ = a₀rⁿ · Triangular n(n + 1)/2 · Star 6n(n − 1) + 1Worked 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.
Frequently asked questions
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.