Number Pattern Finder

Enter a comma-separated list of numbers to detect the pattern, predict the next 5 terms, and get the nth-term formula.

Examples:

What Are Number Patterns?

A number pattern is any sequence of numbers governed by a consistent rule. Recognising patterns is a core skill in mathematics — it underpins algebra, calculus, and even data science. When you spot the rule, you can predict every future term and write a compact formula that describes the entire infinite sequence.

Patterns appear everywhere: the way bacteria populations double, the way distances between branches on a fern are proportional, or the way musical notes relate to each other. This tool automatically checks your sequence against the most common mathematical patterns so you get an instant identification.

Types of Number Sequences

The most important families of sequences you will encounter are:

Arithmetic Sequences

Each term is obtained by adding a fixed number called the common difference (d) to the previous term. Example: 5, 10, 15, 20 has d = 5. The nth term is a + (n − 1)d. These sequences model uniform growth such as saving a fixed amount of money each month.

Geometric Sequences

Each term is multiplied by a fixed number called the common ratio (r). Example: 2, 6, 18, 54 has r = 3. The nth term is a × r^(n − 1). These describe exponential growth (compound interest, population growth) and exponential decay (radioactive half-life).

Fibonacci-Like Sequences

Each term is the sum of the two terms before it. The classic Fibonacci sequence is 1, 1, 2, 3, 5, 8, 13, 21... The ratio of successive terms converges to the golden ratio φ ≈ 1.618. Fibonacci patterns appear in flower petal counts, spiral shells, and galaxy arms.

Figurate Numbers

Square numbers (1, 4, 9, 16, 25…) represent dots arranged in squares; the nth term is n². Triangular numbers (1, 3, 6, 10, 15…) represent dots in triangular arrays; the nth term is n(n+1)/2. Cube numbers (1, 8, 27, 64…) extend the concept to three dimensions; nth term is n³.

How to Find a Pattern — Step-by-Step

  1. Calculate first differences — subtract each term from the next. If all differences are equal, it's arithmetic.
  2. Calculate ratios — divide each term by the previous. If all ratios are equal, it's geometric.
  3. Check for Fibonacci rule — does term 3 = term 1 + term 2, and so on for every subsequent term?
  4. Check figurate formulas — do the numbers fit n², n³, or n(n+1)/2?
  5. Calculate second differences — if second differences are constant, the sequence follows a quadratic formula.
  6. Look for prime numbers — if all terms are prime, it may be a sub-sequence of primes.

Common Number Patterns

Pattern Type Example nth Term Key Property
Arithmetic3, 7, 11, 15a+(n−1)dConstant difference d
Geometric2, 6, 18, 54a·r^(n−1)Constant ratio r
Square numbers1, 4, 9, 16, 25n²Perfect squares
Cube numbers1, 8, 27, 64n³Perfect cubes
Triangular1, 3, 6, 10, 15n(n+1)/2Triangular dot arrays
Fibonacci1, 1, 2, 3, 5, 8T(n−1)+T(n−2)Sum of two prior terms
Powers of 21, 2, 4, 8, 162^(n−1)Doubling sequence

Frequently Asked Questions

A number pattern is a sequence of numbers that follows a predictable rule or formula. Identifying the rule lets you predict every future term and express the entire sequence with a compact algebraic formula. Common patterns include arithmetic (constant difference), geometric (constant ratio), and figurate sequences like square or triangular numbers.
Subtract each term from the next. If every difference is the same, the sequence is arithmetic. For example, 5, 9, 13, 17 has differences of 4 throughout. The general term formula is a + (n − 1)d, where a is the first term and d is the common difference.
Divide each term by the previous one. If every ratio is identical, the sequence is geometric. For example, 3, 12, 48, 192 gives ratios of 4 throughout. The nth term is a × r^(n − 1). Geometric sequences with |r| < 1 converge towards zero; those with |r| > 1 grow without bound.
The Fibonacci sequence starts 1, 1, 2, 3, 5, 8, 13, 21, 34, 55... where each term equals the sum of the two before it. It was introduced to Europe by Leonardo Fibonacci in 1202, though it was known in India centuries earlier. The ratio of consecutive terms converges to the golden ratio φ ≈ 1.618, and the sequence appears in flower petal counts, nautilus shells, and galaxy spirals.
Square numbers are perfect squares — numbers that can form a square array of dots: 1, 4, 9, 16, 25, 36… The nth square number is n². Triangular numbers count dots arranged in equilateral triangular patterns: 1, 3, 6, 10, 15, 21… The nth triangular number is n(n + 1)/2. Note that every square number is the sum of two consecutive triangular numbers.
For arithmetic sequences use T(n) = a + (n − 1)d. For geometric sequences use T(n) = a × r^(n − 1). For quadratic sequences, check whether the second differences are constant; you can then fit T(n) = An² + Bn + C. For Fibonacci-like sequences, no simple closed form exists (Binet's formula requires irrational numbers). The tool will display the formula whenever one can be detected.
If none of the standard patterns match, the tool shows first differences, second differences, and term-to-term ratios. Constant first differences → arithmetic. Constant second differences → quadratic. Constant ratios → geometric. Examining these clues helps you identify the underlying rule manually or spot a custom pattern not covered by standard categories.

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