What is an Arithmetic Progression?

Arithmetic Progression (AP)

An arithmetic progression is a sequence of numbers in which the difference between any two consecutive terms is always the same constant value.

Arithmetic progressions (APs) are one of the most fundamental patterns in mathematics. You encounter them every day — house numbers on a street, seat numbers in a cinema, or the marks on a ruler are all examples of arithmetic progressions.

The key idea is constant difference. Each term is obtained from the previous one by adding (or subtracting) the same fixed number.

Example

Spotting an AP:

3, 7, 11, 15, 19, ... → difference = +4 each time ✓
20, 17, 14, 11, 8, ... → difference = −3 each time ✓
2, 4, 8, 16, 32, ... → differences are 2, 4, 8, 16 — NOT constant ✗ (this is a geometric sequence)

Analogy

Think of climbing a staircase where every step is exactly the same height. Each step up (or down) represents the common difference. As long as all steps are equal, you're walking through an arithmetic progression!

Key Vocabulary: Terms, First Term, and Common Difference

To work with arithmetic progressions confidently, you need to know the standard notation used in mathematics.

First Term (a)

The first term of an arithmetic progression, usually denoted by the letter a (sometimes $a_{1}$ ).

Common Difference (d)

The fixed value added to each term to get the next term, denoted by d. It can be positive, negative, or zero.

nth Term

The term at position n in the sequence, written as $a_{n}$ or $T_{n}$ .

Here is a labelled example using the sequence 5, 9, 13, 17, 21, ...

$a = 5$ (first term)
$d = 9 - 5 = 4$ (common difference)
The 5th term = 21

Note

To find the common difference d, subtract any term from the term that follows it:
$d = a_{2} - a_{1} = a_{3} - a_{2} = a_{n + 1} - a_{n}$
Always check more than one pair of consecutive terms to confirm the sequence is truly arithmetic.

Warning

The common difference d can be negative! A sequence that is decreasing (going down) is still an AP — it just has a negative common difference. For example: 100, 90, 80, 70, ... has $d = - 10$ .

Free preview

10 more sections in this topic

Pick this up in your Library: it holds the whole topic, notes, cheatsheet and questions.