medium difficulty6 min

Arithmetic progression

An arithmetic progression, or AP, is a sequence in which each term after the first is obtained by adding the same fixed number every time.

In one line: Think of an AP as a staircase with equal steps.

Practice This Concept

Understand it

In an AP, the change from one term to the next stays constant throughout the sequence. That fixed change is called the common difference. CBSE questions usually test whether students can identify the pattern, write the next terms, and use formulas correctly after spotting this regular step.

Example

The sequence 4, 9, 14, 19, ... is an AP because each term increases by 5.

Formula and rule

If a sequence has terms a, a + d, a + 2d, a + 3d, ... then it is an AP, where d is the common difference.

See it worked out

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

Worked example

1For 2, 6, 10, 14, ... the differences are 4, 4, and 4.
2Since the difference is constant, the sequence is an AP with first term 2 and common difference 4.

Method you can reuse

1Write the terms in order.
2Subtract each term from the next one.
3If all differences are equal, the sequence is an AP.
4If the difference changes, it is not an AP.

Sketch it in your answer

Write the sequence in a horizontal line and mark the gap between consecutive terms to compare the differences clearly.

Check yourself

Answer the question in your head before opening the answer.

Try it yourself

How do you know that a sequence is an arithmetic progression?

Think first, then show answer

A sequence is an arithmetic progression when the difference between every pair of consecutive terms is the same throughout the list. That fixed difference should not change from one step to the next.

Common wrong answer

A common wrong answer is to say a sequence is AP only because it is increasing.

Why it loses marks: The increase must be by equal steps.

Don't mix this up

Students sometimes treat any increasing sequence as an AP, even when the differences are not equal.

Exam tip

Always check the difference between consecutive terms before calling a sequence an AP.

Write it in the exam

Each point is worth about one mark. Write point 1 for a 1-mark question and keep adding points as the marks go up.

How CBSE asks it

Students may be asked to identify whether a given sequence is an AP, complete missing terms, or use the sequence as a starting point for formulas.

1
Enough for 1 mark
An arithmetic progression, or AP, is a sequence in which each term after the first is obtained by adding the same fixed number every time.
2
Add for 2 marks
Example: The sequence 4, 9, 14, 19, ... is an AP because each term increases by 5.
3
Add for 3 marks
In an AP, the change from one term to the next stays constant throughout the sequence.

MCQ Quiz

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