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Arithmetic Progression — Definition

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

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Student-friendly explanation

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.

How to write this in exams

  1. 1

    Start with the exact idea

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

  2. 2

    Then show how to use it

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

  3. 3

    Add one concrete example

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

  4. 4

    Avoid this incomplete answer

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

Definition

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

Example

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

Rule to remember

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

Memory hook

AP means same difference every step.

Examples and method

Worked example

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

Method to apply

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

Diagram support

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

How CBSE asks it

Students may be asked to verify whether a list is an AP, find d, find the nth term, or find a missing term.

Avoid common mistakes

Common confusion

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

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.

Exam tip

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

Quick check

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

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

Study the arithmetic progression — definition diagram carefully

Use the labelled diagram to keep arithmetic progression — definition clear in short answers and revision.

What this diagram makes clear

This diagram keeps the labels and direction of arithmetic progression — definition in the right order.

Where this helps in exams

Use this for labelled diagram work and short exam answers on arithmetic progression — definition.

Revision cue

Revise arithmetic progression — definition through the labels before writing the answer.

Answer writing and exam use

1-mark answer

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

2-mark answer

An arithmetic progression is a sequence in which the difference between every two consecutive terms is constant. For an AP, d = next term - previous term and an = a + (n - 1)d. 7, 11, 15, 19 is an AP with first term a = 7 and common difference d = 4.

3-mark answer

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. For an AP, d = next term - previous term and an = a + (n - 1)d. Find the 15th term of 3, 8, 13, 18. Here a = 3 and d = 5. a15 = 3 + (15 - 1)5 = 3 + 70 = 73. Students may be asked to verify whether a list is an AP, find d, find the nth term, or find a missing term. For 20, 17, 14, 11, some students write d = 3, but the correct common difference is -3 because each term is decreasing.
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