Finding the simplest matching pattern...
Enter at least three terms to identify a pattern and predict what may come next.
What Is a Number Pattern Finder?
A number pattern finder compares the transitions in a finite sequence with common mathematical rules. It can recognize constant changes, constant ratios, previous-two-term sums, polynomial finite differences, alternating subsequences, and multiply-add recurrences.
A finite list never proves one unique continuation. Infinitely many rules can match the same terms. This calculator therefore chooses the simplest supported rule that reproduces every supplied value and shows alternative matches when they exist.
How the Sequence Finder Tests a Pattern
The recognizer moves from simpler rules to more structured recurrences while validating the complete input.
Simplicity ranking
Straightforward rules outrank more complicated models when both match.
Full validation
Every supplied transition must agree with the detected rule.
Ambiguity warning
Predictions describe one plausible rule, not certain author intent.
How to Find the Next Number in a Sequence
Start by subtracting consecutive terms. Equal first differences indicate an arithmetic sequence. If differences are not equal, divide consecutive nonzero terms and check for a constant geometric ratio.
Next test whether each term is the sum of the previous two. When none of these rules fits, build second, third, and fourth differences, split odd and even positions, and test a multiply-add recurrence.
Always test a proposed rule against the entire list. A relationship that explains only the last pair is weak evidence and may fail immediately on earlier terms.
Arithmetic, Geometric, and Fibonacci Patterns
An arithmetic sequence adds a common difference d. Its nth term is a₁+(n−1)d. A geometric sequence multiplies by a common ratio r and has nth term a₁rⁿ⁻¹.
A Fibonacci-style sequence adds the previous two terms. Starting values need not be 0 and 1; any two real starting values can define the same recurrence.
Zeros need special care in geometric detection because division by zero is undefined. The calculator validates geometric values recursively instead of trusting isolated ratios.
Finite Differences and Polynomial Sequences
Equal second differences indicate values generated by a quadratic polynomial in the term index. Equal third differences correspond to a cubic pattern, while equal fourth differences correspond to a quartic pattern.
Prediction extends the deepest constant row and accumulates upward through the difference triangle. This keeps calculations consistent without fitting a high-degree polynomial through every arbitrary list.
The tool intentionally stops at fourth differences. Although a polynomial of sufficiently high degree can interpolate any finite list, presenting that automatic fit as a meaningful discovered pattern would be misleading.
Common Sequence Pattern Rules
| Pattern | Recognition test | Next-term rule |
|---|---|---|
| Arithmetic | Equal first differences | Add d |
| Geometric | Equal ratios | Multiply by r |
| Fibonacci-style | Previous two terms sum | Add the last two |
| Quadratic | Equal second differences | Extend differences |
| Alternating | Odd/even positions have rules | Extend matching parity |
| Affine recurrence | aₙ = maₙ₋₁+b fits | Multiply then add |
Number Sequence Finder FAQ
How do I find the pattern in a number sequence?
Check differences, ratios, sums of previous terms, and finite differences in that order. Validate any proposed rule against every supplied term.
What is the difference between arithmetic and geometric sequences?
Arithmetic sequences add a constant difference. Geometric sequences multiply by a constant ratio.
How do you predict the next Fibonacci number?
Add the final two known terms. Repeat the same recurrence using each newly predicted term.
What do second differences tell you?
Constant second differences indicate a quadratic pattern in the term index. Extending the difference table predicts following values.
Can one number sequence have multiple patterns?
Yes. Any finite list can match many possible formulas. The calculator ranks simple supported matches but cannot know the author's intention.
Why did the calculator not predict a next term?
No supported rule matched every supplied transition. Refusing to guess is more accurate than inventing a high-degree interpolation.
Can sequence terms be negative or decimal numbers?
Yes. Finite negative values, decimals, zero, and scientific notation are accepted.