Types of Number Patterns: Arithmetic, Geometric, Fibonacci & More

Almost every number sequence puzzle you'll meet is built from a small set of pattern types. Learn to recognize these, and most "what comes next" questions become instant. This guide is a reference to the main types of number patterns, arithmetic, geometric, square, triangular, Fibonacci, prime, and alternating, each with a clear example and how to spot it. For the step-by-step method of applying them, pair this with how to solve number sequence puzzles.

Arithmetic sequences (constant difference)

The most common type. Each term differs from the next by the same fixed amount.

• Example: 3, 7, 11, 15, 19 (add 4 each time).
• How to spot it: the differences between terms are all equal.
• Next term: add the common difference once more.

Arithmetic patterns are your first guess on almost any sequence, because they're so common. Our easy pattern puzzles are built mostly from these.

Geometric sequences (constant ratio)

Each term is the previous one multiplied (or divided) by the same number.

• Example: 2, 6, 18, 54 (multiply by 3 each time).
• How to spot it: dividing each term by the previous one gives the same ratio.
• Next term: multiply by the common ratio again.

Geometric sequences grow (or shrink) fast, so if the numbers are ballooning, suspect a ratio rather than a difference.

Square numbers

The squares of the counting numbers: each term is its position multiplied by itself.

• Example: 1, 4, 9, 16, 25, 36 (1², 2², 3², 4², 5², 6²).
• How to spot it: the differences increase by a constant amount (here 3, 5, 7, 9, the odd numbers).
• Next term: square the next position number.

Triangular numbers

The running totals of the counting numbers: 1, then 1+2, then 1+2+3, and so on.

• Example: 1, 3, 6, 10, 15, 21 (formula n(n+1)/2).
• How to spot it: the differences are 2, 3, 4, 5, the counting numbers themselves.
• Next term: add the next counting number.

Cube numbers

The cubes of the counting numbers.

• Example: 1, 8, 27, 64, 125 (1³, 2³, 3³, 4³, 5³).
• How to spot it: rapid growth that isn't geometric; check whether each term is a perfect cube.

The Fibonacci sequence

A famous recursive pattern where each term is the sum of the two before it.

• Example: 1, 1, 2, 3, 5, 8, 13, 21 (each term = sum of previous two).
• How to spot it: no constant difference or ratio, but each term equals the previous two added together.
• Next term: add the last two terms.

The Fibonacci sequence shows up surprisingly often in puzzles, so it's worth memorizing the opening terms.

Prime numbers

The numbers divisible only by 1 and themselves.

• Example: 2, 3, 5, 7, 11, 13, 17 (the primes in order).
• How to spot it: an irregular sequence with no arithmetic or geometric rule, where every term is prime.
• Next term: the next prime number.

Primes are a favorite "trick" pattern because they look irregular until you recognize them.

Alternating and interleaved patterns

Two patterns woven together, taking turns.

• Example: 1, 10, 2, 20, 3, 30 (odd positions 1, 2, 3; even positions 10, 20, 30).
• How to spot it: no single rule fits, but splitting the sequence into odd-positioned and even-positioned terms reveals two simple patterns.
• Next term: continue whichever sub-sequence is next.

These define our medium pattern puzzles, where the "split it in two" technique is essential.

Polynomial and composite patterns

The hardest types combine operations or follow position-based formulas beyond squares.

• Example (polynomial): 2, 6, 12, 20, 30 follows n × (n+1).
• Example (composite): 5, 11, 23, 47 follows "multiply by 2, add 1."
• How to spot it: constant second differences point to a quadratic; if even that fails, try combining two operations.

These power our expert and Einstein pattern puzzles, where you often test several hypotheses before one fits.

A quick recognition table

Pattern type	Example	Tell-tale sign
Arithmetic	3, 7, 11, 15	constant difference
Geometric	2, 6, 18, 54	constant ratio
Square	1, 4, 9, 16	differences 3, 5, 7...
Triangular	1, 3, 6, 10	differences 2, 3, 4...
Cube	1, 8, 27, 64	perfect cubes
Fibonacci	1, 1, 2, 3, 5	sum of previous two
Prime	2, 3, 5, 7, 11	all prime, irregular gaps
Alternating	1, 10, 2, 20	two woven sub-sequences

Put the types to work

Knowing the types is half the battle; the other half is matching them quickly. The more puzzles you solve, the faster you'll recognize each one on sight. Try spotting these types in our pattern puzzles, starting easy and climbing as your eye sharpens. And when a sequence stumps you, the solving method walks you through narrowing it down.