Number Sequence Pattern Solver
Type the start of a number sequence to find its rule and nth-term formula, the next five terms, any term you choose and the table of differences.
About this tool
The solver works out the differences between neighbouring terms. If the first differences are all the same, the sequence is arithmetic (linear) and the nth term is an + b; if the second differences are the same, it is quadratic (an² + bn + c), as in GCSE questions; and if the third differences match, it is cubic. It also spots geometric sequences, where each term is multiplied by the same number, and Fibonacci-style ones, where each term is the sum of the two before.
All the working uses exact fractions, so the formula for the triangular numbers comes out as (1/2)n² + (1/2)n rather than rounded decimals, and the term finder can go as far as the 10,000th term. Give at least four terms for a quadratic and five for a cubic so the pattern can be checked, not just fitted. Any short list of numbers can be continued in many ways, so the tool gives the simplest rule of these types that fits.