What is an Arithmetic Sequence?

Arithmetic Sequence

An arithmetic sequence is an ordered list of numbers where the difference between any two consecutive terms is always the same constant value.

Arithmetic sequences are everywhere in science and everyday life. Think about:

A stack of equal-thickness books (each book adds the same height)
A taxi fare that charges a fixed amount per kilometre
The depth markings on a ruler (0 cm, 1 cm, 2 cm, 3 cm…)

In each case, you are adding the same amount each time you move to the next term. That constant amount is called the common difference.

Common Difference (d)

The common difference is the fixed amount added to each term to get the next term in an arithmetic sequence. It can be positive (sequence increases), negative (sequence decreases), or zero (sequence is constant).

Example

Identifying the common difference

Sequence: 3, 7, 11, 15, 19, …

7 − 3 = 4
11 − 7 = 4
15 − 11 = 4

The common difference $d = 4$ . Each term is 4 more than the previous one.

Sequence: 20, 15, 10, 5, 0, …

15 − 20 = −5
10 − 15 = −5

The common difference $d = - 5$ . This is a decreasing arithmetic sequence.

Warning

Not every number pattern is arithmetic! Check that the difference is constant between ALL consecutive pairs. The sequence 1, 2, 4, 8, 16 looks regular but the differences are 1, 2, 4, 8 — they are not equal, so this is not arithmetic.

The Language and Notation of Sequences

To work with sequences efficiently, mathematicians use a standard notation. Let's break it down:

The terms of a sequence are labelled $a_{1}, a_{2}, a_{3}, \dots, a_{n}$
$a_{1}$ is the first term (also often written as $a$ )
$n$ is the position of a term in the sequence (the term number)
$a_{n}$ is the general term — the value of the term at position $n$
$d$ is the common difference

Analogy

Think of a sequence like seats in a cinema row. Seat 1 ( $a_{1}$ ) is the first seat. Seat 5 ( $a_{5}$ ) is the fifth seat. The row number tells you the position, and each seat is a fixed width apart — that fixed width is like the common difference!

For the arithmetic sequence 5, 8, 11, 14, 17, …:

Position ( $n$ )	1	2	3	4	5
Term ( $a_{n}$ )	5	8	11	14	17

First term: $a_{1} = 5$
Common difference: $d = 3$
Fifth term: $a_{5} = 17$

Note

The common difference is always found by subtracting a term from the term that follows it:
$d = a_{n + 1} - a_{n}$
You can use any two consecutive terms — the result will always be the same in an arithmetic sequence.

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