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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.

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Student-friendly explanation

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.

How to write this in exams

  1. 1

    Start with the exact idea

    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. 2

    Then show how to use it

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

  3. 3

    Add one concrete example

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

  4. 4

    Avoid this incomplete answer

    A common wrong answer is to say a sequence is AP only because it is increasing. That is wrong because the increase must be by equal steps.

Definition

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.

Example

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

Rule to remember

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

Memory hook

Think of an AP as a staircase with equal steps.

Examples and method

Worked example

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

Method to apply

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

Diagram support

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

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.

Avoid common mistakes

Common confusion

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

Common wrong answer

A common wrong answer is to say a sequence is AP only because it is increasing. That is wrong because the increase must be by equal steps.

Exam tip

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

Quick check

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

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.

Answer writing and exam use

1-mark answer

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-mark answer

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. If a sequence has terms a, a + d, a + 2d, a + 3d, ... then it is an AP, where d is the common difference. The sequence 4, 9, 14, 19, ... is an AP because each term increases by 5.

3-mark answer

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. If a sequence has terms a, a + d, a + 2d, a + 3d, ... then it is an AP, where d is the common difference. For 2, 6, 10, 14, ... the differences are 4, 4, and 4. Since the difference is constant, the sequence is an AP with first term 2 and common difference 4. 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. A common wrong answer is to say a sequence is AP only because it is increasing. That is wrong because the increase must be by equal steps.
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