Sequence Calculator

Background.

Paste a list of numbers and this calculator works out which rule generated them, writes that rule down, and continues the sequence for as many terms as you want. It is the reverse of every other sequence page: instead of asking you for a first term and a common difference and handing back a value, it starts from the numbers you can actually see and tells you what they are.

That distinction matters more than it sounds. If you are staring at 2, 5, 8, 11, 14 and wondering what comes next, an arithmetic sequence calculator is no help until you have already spotted that it is arithmetic and worked out that the common difference is 3 — which is the whole job. This page does that step. It reports the family in plain words, gives you the nth-term rule you can write in a homework answer, and generates the continuation.

Four rules are recognised, and the page names all four rather than pretending to be a universal pattern-finder. Arithmetic: every step adds the same amount. Geometric: every step multiplies by the same factor. Quadratic: the gaps between terms themselves grow by a constant amount, which is what perfect squares, triangular numbers and any second-degree polynomial do. Fibonacci-style: each term is the sum of the two before it, from any starting pair — not just 1 and 1. If your numbers fit none of those, the calculator says so and stops. It does not fit a curve to unrecognised data and it does not extrapolate from the last two terms, because a confident wrong answer about what comes next is worse than an honest refusal.

The order the four tests run in is fixed and worth knowing, because short sequences can genuinely satisfy more than one rule. The list 1, 2, 3 is arithmetic with a difference of 1 and also satisfies the Fibonacci rule, since 3 = 1 + 2. Testing arithmetic first means the simpler description wins, which is almost always the one intended. The full order is arithmetic, then geometric, then quadratic, then Fibonacci-style.

A minimum of four terms is required. With three, the quadratic test has only one second difference to look at and the Fibonacci test only one sum to check, so almost any three numbers satisfy several rules at once and the answer would be close to meaningless. Four is the smallest count at which the two-step families each get two independent checks. Five or more is better, and the page says so rather than quietly giving you a shaky verdict.

One technical detail that changes results and is therefore stated openly: your numbers are read directly from the text you typed into an exact decimal engine, never through binary floating point. That is why 0.1, 0.2, 0.3, 0.4 is correctly identified as arithmetic with a common difference of exactly 0.1. A calculator that goes through ordinary floating-point arithmetic computes 0.3 − 0.2 as 0.09999999999999998, finds the differences unequal, and misses the pattern entirely. Differences are therefore compared exactly, with no tolerance at all. Ratios cannot be compared exactly — one third has no finite decimal expansion — so the geometric test uses a relative tolerance of one part in a trillion, which is why 81, 27, 9, 3 is correctly recognised even though its ratio never terminates.

Once you know the family, the dedicated pages go deeper: arithmetic sequence for the nth term and partial sums, geometric sequence for the convergence test on infinite series, and fibonacci for exact large-index values.

What is sequence calculator?

A number sequence is an ordered list where position matters — the first term, the second term, and so on — and a sequence rule is a description of how each term is produced. This calculator tests an entered list against four standard families. An arithmetic sequence has a constant difference between consecutive terms, so aₙ = a₁ + d(n − 1); OpenStax College Algebra 2e §9.2 defines it as the case where 'the difference between any two consecutive terms is a constant'. A geometric sequence has a constant ratio, aₙ = a₁r^(n−1), which §9.3 describes as increasing or decreasing 'by a constant factor called the common ratio'. A quadratic sequence has a constant second difference and is generated by a second-degree polynomial An² + Bn + C; the perfect squares and the triangular numbers are the two most familiar examples. A Fibonacci-style sequence obeys the two-step recurrence aₙ = aₙ₋₁ + aₙ₋₂ from any starting pair, generalising the classic Fibonacci numbers. These four cover the overwhelming majority of sequences that appear in school and competition problems; sequences built from other rules — primes, factorials, digit patterns, or anything requiring cleverness rather than arithmetic — are outside this page's scope and it declines them explicitly.

How to use this calculator.

1. 01Type or paste your numbers in order, separated by commas or spaces. At least four are needed; five or more gives a firmer answer.
2. 02Set how many further terms you want generated, from 1 to 50.
3. 03Read the pattern line first — it names the family and the parameter, for example 'each term is 3 more than the one before it'.
4. 04Use the nth-term rule for homework and the continued sequence to check your own working.
5. 05If the calculator refuses, your numbers do not fit any of the four recognised rules. Double-check for a typo or a term entered out of order before assuming the pattern is exotic.

The formula.

The calculator takes your terms and runs four structural tests in a fixed order. First it takes every first difference aₖ − aₖ₋₁ and checks whether they are all identical; if so the sequence is arithmetic and the rule is aₙ = a₁ + d(n − 1). Next it takes every ratio aₖ ⁄ aₖ₋₁ — skipping this test entirely if any term is zero, since a geometric sequence never contains one — and checks whether those agree; if so the rule is aₙ = a₁r^(n−1). Third it takes second differences, the changes in the first differences, and checks those for constancy. A constant second difference Δ² means the terms come from a quadratic, and the coefficients follow from a₁, the first difference Δ₁ and Δ² as A = Δ²/2, B = Δ₁ − 3Δ²/2 and C = a₁ − Δ₁ + Δ². You can check that derivation on the squares 1, 4, 9, 16: a₁ = 1, Δ₁ = 3, Δ² = 2, giving A = 1, B = 3 − 3 = 0 and C = 1 − 3 + 2 = 0, so the rule reduces to aₙ = n² as it should. Last it checks whether every term from the third onward equals the sum of the two before it. The order matters because 1, 2, 3 satisfies both the arithmetic and the Fibonacci tests, and the simpler description is almost always the intended one. On precision: your numbers are parsed straight from the characters you typed into a decimal engine, so differences are compared with no tolerance whatsoever — 0.1, 0.2, 0.3 has an exactly constant difference here, where floating-point arithmetic would see 0.09999999999999998 and reject it. Ratios get a relative tolerance of 1e-12 because a ratio such as one third has no finite decimal form. Nothing is rounded during detection; the three numeric outputs are rounded once at the end to 10 decimal places, and the text outputs show 12 significant figures, which is why a repeating ratio prints as 0.333333333333 rather than as a fraction.

A worked example.

Frequently asked questions.

References& sources.

1. [1]OpenStax (Rice University), College Algebra 2e, §9.2 'Arithmetic Sequences' — 'The difference between any two consecutive terms is a constant. This constant is called the common difference', and aₙ = a₁ + d(n − 1). Retrieved 2026-07-29; open access.
2. [2]OpenStax (Rice University), College Algebra 2e, §9.3 'Geometric Sequences' — 'Each term of a geometric sequence increases or decreases by a constant factor called the common ratio', and aₙ = a₁r^(n−1). Retrieved 2026-07-29; open access.
3. [3]Wolfram MathWorld, 'Fibonacci Number' — equation 1, the two-step recurrence Fₙ = Fₙ₋₁ + Fₙ₋₂ that this page's fourth family generalises to arbitrary starting pairs. Retrieved 2026-07-29; open access.
4. [4]Encyclopedia of Mathematics (European Mathematical Society), 'Arithmetic progression' — independent check on the arithmetic test: the characteristic property aₙ = (aₙ₊₁ + aₙ₋₁)/2, every term being the mean of its neighbours. Applied to 2, 5, 8, 11, 14 it reaches the same verdict as the constant-difference test. Retrieved 2026-07-29; open access.
5. [5]OpenStax (Rice University), Calculus Volume 1, §5.1 'Approximating Areas', rule 'Sums and Powers of Integers' — Σ i = n(n+1)/2 and Σ i² = n(n+1)(2n+1)/6, the closed forms behind the triangular and square-number examples used for the quadratic family. Retrieved 2026-07-29; open access.