How do you find the rule for a sequence?

To find the rule, look at how you get from one term to the next. Check if the sequence adds or subtracts the same number each time (arithmetic), multiplies by the same number each time (geometric), or follows another clear pattern. Then describe that pattern in words or as a formula.

Is a sequence a function?

Yes. A sequence can be viewed as a function whose input is usually a counting number (1, 2, 3, ...) and whose output is the term in that position. For example, if f(3) = 15, then the 3rd term of the sequence is 15.

What is the difference between a sequence and a pattern?

In early grades, the words are often used in similar ways, but a sequence usually means a pattern written as an ordered list of terms. A pattern might appear in shapes, tables, or pictures, while a sequence focuses on the ordered list of terms that follow the pattern.

What Is a Sequence in Math? Definition and Examples

Q: What is a sequence in math?

A sequence in math is an ordered list of numbers or objects that follow a specific rule or pattern. Each item in the list is called a term, and the position of each term in the list also matters.

Kid-friendly explanation

Think of a sequence like a list of steps that follow a rule. For example, counting 1, 2, 3, 4, 5 is a sequence that goes up by 1 each time. Lining up blocks as 2 blocks, then 4, then 6 is a sequence that goes up by 2. A sequence is any list that follows a clear pattern!

Worked examples

Example 1 (younger grades): finding the next terms in a simple sequence

Look at this sequence: 5, 10, 15, 20, ? , ?

First, find the pattern. From 5 to 10, the sequence adds 5. From 10 to 15, it also adds 5. From 15 to 20, it adds 5 again. So the rule is "add 5 each time." The next term is 20 + 5 = 25. The term after that is 25 + 5 = 30. So the completed sequence is 5, 10, 15, 20, 25, 30.

Example 2 (upper grades): geometric sequence and function rule

Consider this sequence: 3, 6, 12, 24, 48, ...

First, look for the pattern between terms. From 3 to 6, multiply by 2. From 6 to 12, multiply by 2. From 12 to 24, multiply by 2. From 24 to 48, multiply by 2. So this is a geometric sequence, where each term is multiplied by 2 to get the next term.

Number the terms using n = 1, 2, 3, 4, ... . The 1st term is 3, the 2nd is 6, the 3rd is 12, and so on. A geometric sequence like this can be written as a function. One explicit rule (a formula that gives any term directly) is:

f(n) = 3 × 2^{n - 1}, where n = 1, 2, 3, ...

Check the 4th term using the rule: f(4) = 3 × 2^{4 - 1} = 3 × 2³ = 3 × 8 = 24, which matches the sequence.

Going deeper with sequences

In early grades, a sequence often means the count sequence (like 0, 1, 2, 3, ...) or simple growing patterns with numbers or shapes. Students describe what comes next and explain the rule, such as "add 2 each time" or "each new shape has 1 more square."

Later, students learn special types of sequences, such as arithmetic sequences (add or subtract the same amount each time) and geometric sequences (multiply by the same amount each time). They may write rules in words, as formulas, or as functions, and calculate the sum of a finite sequence.

Sequences also appear in geometry. A "sequence of transformations" is a set of moves, in a specific order, such as translations (slides), rotations (turns), reflections (flips), and dilations (resizing). Using a sequence of transformations, students can show when two shapes are congruent (same size and shape) or similar (same shape, proportional size).

Common mistakes with sequences

Ignoring the order of terms. A student might think {2, 4, 6} and {6, 4, 2} are the same sequence because they use the same numbers. Correction: a sequence is ordered, so changing the order creates a different sequence.
Assuming every pattern is "add the same number." For the sequence 2, 4, 8, 16, a student might say "add 2" because 2 + 2 = 4, then incorrectly continue 4 + 2 = 6, 6 + 2 = 8, and so on. Correction: check each step; here the pattern is "multiply by 2," so the correct next term after 16 is 32.
Mixing up the term number and the term value. A student might say the 3rd term of 5, 10, 15, 20 is 20, because 3 + 17 = 20. Correction: the 3rd term is the third item in the list, which is 15; the term number (position) is not the same as the term's value.

How sequences show up in everyday life

Sequences appear in many daily routines. The days of the week form a sequence in a fixed order. The numbers on a calendar, the levels in a video game, or the pattern of chairs in rows all follow sequences. Even a set of dance steps or moves in a sport drill is a sequence that must be done in the right order.

Related math terms

What is a sequence in math?