medium difficulty5 min

Arithmetic Progression — Definition

An arithmetic progression is a sequence in which the difference between every two consecutive terms is constant.

In one line: AP means same difference every step.

Understand it

In an AP, we repeatedly add the same number called the common difference d. If the first term is a, then the nth term is found by an = a + (n - 1)d. The difference may be positive, negative, or zero.

Example

7, 11, 15, 19 is an AP with first term a = 7 and common difference d = 4.

Formula and rule

For an AP, d = next term - previous term and an = a + (n - 1)d.

See it worked out

Follow the example line by line, then use the same method on new questions.

Worked example

1Find the 15th term of 3, 8, 13, 18.
2Here a = 3 and d = 5. a15 = 3 + (15 - 1)5 = 3 + 70 = 73.

Method you can reuse

1Find consecutive differences.
2Confirm they are equal.
3Identify a and d.
4Use an = a + (n - 1)d for the required term.
5Check the sign of d.

Visual tip

A number-line jump diagram can show equal jumps of +d or -d between terms.

Check yourself

Answer the question in your head before opening the answer.

Try it yourself

Is 20, 17, 14, 11 an AP? Give the common difference.

Think first, then show answer

Yes, it is an AP because each term decreases by 3, so the common difference is d = -3.

Common wrong answer

For 20, 17, 14, 11, some students write d = 3, but the correct common difference is -3 because each term is decreasing.

Don't mix this up

Students sometimes compare non-consecutive terms or use only one pair of terms and conclude too quickly.

Exam tip

Check at least two consecutive differences before calling a sequence an AP.