How to use the Sequence Calculator
- Type the terms you know, separated by commas or spaces (fractions like 1/2 work). Four or more terms make the rule certain.
- Read the type of sequence, its nth term formula and the term-to-term rule, with the differences or ratios that prove it.
- Find any term, the sum of any range of terms, or whether a number is in the sequence. The other modes take a first term with d or r, or two known terms.
What does this tool do?
Most sequences in school and everyday problems follow one of a few rules. The calculator checks them in order: a constant difference (arithmetic), a constant ratio (geometric), a constant second difference (quadratic, an² + bn + c), a constant third difference (cubic), each term being the sum of the two before (Fibonacci-type), and “multiply then add” sequences such as 2, 5, 11, 23 (×2 + 1).
Coefficients are shown as exact fractions, so 2, 3.5, 5, 6.5 gives aₙ = (3/2)n + 1/2 rather than a rounded decimal. The working panel shows the differences or ratios, which is the method GCSE and other exams expect for finding a quadratic nth term.
Once the rule is known you can jump to term 1,000, add up terms 20 to 50, get the sum to infinity of a converging geometric sequence, or check whether a value such as 399 appears and at which position. A table lists the terms with a running total, and you can download the first 1,000 terms as a CSV.
Why use it?
- Finds the rule from the terms — not just from a₁ and d.
- Arithmetic, geometric, quadratic, cubic, Fibonacci-type and recursive rules.
- Exact fractions in formulas, plus the differences that prove them.
- Any term, any range sum, sum to infinity and “is this number a term?”
- Arithmetic or geometric sequence from just two known terms.
Use cases
- Checking nth-term homework, including quadratic sequences at GCSE.
- Finding the next number in a pattern or aptitude-test sequence.
- Savings plans that grow by a fixed amount (arithmetic) or percentage (geometric).
- Seating rows, stacked objects and other counting problems with a regular pattern.
Example: a quadratic sequence
Terms 4, 7, 12, 19, 28. The first differences are 3, 5, 7, 9 and the second differences are all 2, so the nth term has the form an² + bn + c with a = 2 ÷ 2 = 1. Subtracting n² (1, 4, 9, 16, 25) leaves 3, 3, 3, 3, 3, so the formula is aₙ = n² + 3 and the next terms are 39, 52 and 67.
Common mistakes
- Writing the difference as the nth term: for 3, 7, 11, 15 the rule isn't 4n but 4n − 1, because the zeroth term would be −1.
- Halving the second difference twice, or forgetting to halve it, when finding a in an² + bn + c.
- Assuming a pattern from three terms. 1, 2, 4 could be doubling (8 next) or adding 1, 2, 3 (7 next) — add a fourth term.
- Using the infinite-sum formula when |r| ≥ 1: the sum keeps growing and has no finite value.
The formula
Privacy
The calculation happens instantly in your browser. The numbers you enter are not sent to our servers or saved. There's no account to create and nothing to install.
Frequently asked questions
How do I find the nth term of a sequence?
Look at the differences between terms. If they're all the same, the nth term is dn + (a₁ − d). If the ratios are the same, it's a₁·rⁿ⁻¹. If the second differences are constant, it's quadratic: an² + bn + c with a equal to half the second difference. Paste your terms and the calculator shows each step.
What's the next number in the sequence?
Type the terms you have; once a rule is found, the next five terms are listed straight away along with the formula that produces them.
What's the difference between arithmetic and geometric sequences?
An arithmetic sequence adds the same amount each time (5, 8, 11…); a geometric sequence multiplies by the same amount (5, 10, 20…). Arithmetic sequences grow steadily; geometric ones grow (or shrink) faster and faster.
Can I find a sequence from two terms?
Yes, if you know the type. For an arithmetic sequence with a₃ = 11 and a₇ = 27, d = (27 − 11) ÷ (7 − 3) = 4 and a₁ = 3. Use the “Two terms” mode; for geometric sequences it also tells you when a negative ratio works too.
What is the difference between a sequence and a series?
A sequence is the list of terms; a series is their sum. This calculator gives both for its patterns; the Series Calculator adds up any expression in n.
Last reviewed by the M2Toolkit team.