🔢Number Sequence Calculator

About this tool

Type at least three terms of a sequence, separated by commas or spaces, and choose n. The calculator identifies the pattern, gives its formula, lists the next five terms, works out the nth term and the sum of the first n terms, and shows the sum to infinity for a converging geometric series.

How to find the rule for a sequence

Start by looking at the differences between neighbouring terms. If the difference is always the same, the sequence is arithmetic. 3, 7, 11, 15 goes up by 4 each time, so the nth term is 4n − 1, the 20th term is 79 and the sum of the first 20 terms is 20 ÷ 2 × (3 + 79) = 820. If the ratio between terms is constant instead, the sequence is geometric: 2, 6, 18, 54 multiplies by 3, so aₙ = 2 × 3ⁿ⁻¹. A geometric series whose ratio is between −1 and 1 has a finite sum to infinity: 8 + 4 + 2 + 1 + … = 16. If the first differences change but the second differences are constant, the sequence is quadratic, like the square numbers 1, 4, 9, 16 with aₙ = n². If each term is the sum of the two before it, it's Fibonacci-like: 1, 1, 2, 3, 5, 8, with 55 as the 10th term. Honest limits: any finite list can be continued by infinitely many rules, so this tool finds the simplest standard pattern. Puzzles built on primes, alternating rules or digit tricks won't be detected; add more terms if the result looks wrong. The prime number tool lists primes and the exponent calculator works with powers.

Frequently asked questions