Number Sequence Examples: Common Patterns and Special Types

You probably use number sequences every day, even if you don’t realize it. Whether you’re adding the same amount over and over or multiplying, these patterns help you spot order and solve problems a bit faster. A number sequence is just a list of numbers that follows a clear rule—like adding, multiplying, or combining past terms—so you can predict the next numbers with a bit of confidence.

This post walks through common sequence types and gives you practical examples you can use right away. You’ll see arithmetic and geometric sequences, triangular, square, and cube patterns, Fibonacci, and some trickier mixed or arithmetico-geometric cases.

Expect short examples that help you spot patterns and practice finding missing terms.

Essential Number Sequence Examples and Their Patterns

Let’s look at three common types of sequences, each with clear rules and examples.

Each type shows how to find the next terms and how to write the nth term.

Arithmetic Sequence Examples and Common Difference

An arithmetic sequence always adds the same value each time. For example: 2, 5, 8, 11, 14. Here, the common difference is d = 3.

Just keep adding 3 to the last term to continue.

Use the nth term formula: a_n = a_1 + (n − 1)d. If a_1 = 2 and d = 3, then a_10 = 2 + 9×3 = 29.

That lets you jump straight to the 10th term without listing every term.

Common differences can be negative or even zero. For example: 20, 17, 14, 11 has d = −3.

If d = 0, the sequence stays constant: 5, 5, 5, 5.

You can check your work by subtracting consecutive terms to see if the difference stays the same.

Geometric Sequence Examples and Common Ratio

A geometric sequence multiplies by the same value each time. Like: 3, 6, 12, 24, 48. Here, the common ratio is r = 2.

Just multiply the last term by 2 to keep going.

The nth term formula is a_n = a_1 × r^(n−1). So, with a_1 = 3 and r = 2, a_6 = 3 × 2^5 = 96.

That’s a pretty quick way to get to distant terms.

Ratios can be fractions or even negative numbers. For instance: 16, 8, 4, 2 uses r = 1/2.

Another example: 5, −10, 20, −40 has r = −2.

If r = 0, the sequence looks like 5, 0, 0, 0.

Check by dividing a term by the one before it to confirm the ratio.

Fibonacci Sequence Illustrated with Real Examples

The Fibonacci sequence starts with 0 and 1. Each term is the sum of the two before it: 0, 1, 1, 2, 3, 5, 8, 13, 21.

Just add the last two terms (13 + 21 = 34) to keep building.

You’ll see Fibonacci numbers in growth patterns—like pairs, petals, or steps—where each stage depends on the two before it.

To find a term, just keep adding earlier pairs until you reach the spot you want.

There’s no fixed difference or ratio for Fibonacci numbers. The rule is a_n = a_{n−1} + a_{n−2}.

If you need the 12th term, keep building the list up or use a formula from the recurrence.

Special and Advanced Number Sequence Examples

These examples show how specific rules create recognizable patterns you can spot and test. You’ll see formulas, quick checks, and tips for identifying each sequence by their differences or rules.

Square Numbers and Square Number Sequence

Square numbers are n² where n is a natural number. The sequence goes 1, 4, 9, 16, 25, … and grows faster than anything linear.

Just use an = n² to get any term you need.

You can check a list by taking the square root of each number—if they come out as whole numbers in order, you’re looking at square numbers.

Squares show up in all sorts of problems and patterns, especially with shapes (think of dots making squares).

You’ll also find them as sums of odd numbers: 1 + 3 + 5 + … + (2n−1) = n².

Square numbers can go on forever or stop, depending on the problem.

If you look at differences between terms—3, 5, 7, 9, …—those odd numbers are the second pattern behind squares.

Triangular Numbers and Triangular Number Sequence

Triangular numbers count dots that make equilateral triangles: 1, 3, 6, 10, 15, …

The nth triangular number is n(n+1)/2.

This formula lets you find any term without writing out the whole list.

To check if a number t is triangular, solve n² + n − 2t = 0; if n comes out as a positive whole number, t is triangular.

Triangular numbers pop up in series and combinatorics, like in the formula C(n+1,2).

The differences between triangular numbers are just the natural numbers: 2, 3, 4, 5, …

That makes them pretty easy to spot.

You’ll often see triangular numbers mixed with square and cube numbers in contest problems.

Quadratic Sequence Examples and Second Difference

Quadratic sequences follow an² + bn + c, so their second differences stay constant.

If you list out the terms and work out the first and second differences, a fixed second difference means you’ve got a quadratic sequence.

For example: 1, 2, 4, 7, 11, … has first differences 1, 2, 3, 4 and second differences all 1.

That tells you the general term is quadratic.

You can use a table or solve three equations from three terms to find a, b, and c.

Quadratic sequences show up as triangular or square-related sequences, or as patterns like n³ + 1 in some lists.

Spotting a constant second difference saves you time. It helps you separate quadratic sequences from arithmetic (constant first difference) and geometric (constant ratio) types.

Harmonic, Prime, and Other Unique Sequence Types

Harmonic sequences pop up when you take the reciprocals of an arithmetic sequence: 1, 1/2, 1/3, 1/4, and so on.

They don’t usually fit neat polynomial rules, but if you flip the terms, you’ll spot a linear pattern.

Prime numbers start with 2, 3, 5, 7, 11, and keep going.

Nobody’s found a simple formula that spits out every prime, so you have to test each number for primality instead.

You’ll also run into cube numbers (n³), odd numbers (2n−1), and some mixed sequences.

Some sequences get built from recurrence relations, like the Fibonacci rule: an = an−1 + an−2.

When you’re trying to figure out what kind of sequence you’re dealing with, try these checks:

• Look at the differences or ratios (arithmetic versus geometric).
• Take reciprocals if you think it’s harmonic.
• Use polynomial-degree tests by checking successive differences (if the k-th difference is constant, it’s degree k).

These tricks can help you sort out whether a sequence is finite or infinite, and sometimes they make finding missing terms way easier.