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Choosing whether a sequence is an AP

Choosing whether a sequence is an AP means checking if the differences between consecutive terms are all equal.

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Student-friendly explanation

This skill is essential because students must not assume every regular-looking list is an AP. Some sequences increase unevenly, while others may follow a different pattern entirely. The safest method is to compare consecutive differences carefully and consistently.

How to write this in exams

  1. 1

    Start with the exact idea

    Choosing whether a sequence is an AP means checking if the differences between consecutive terms are all equal.

  2. 2

    Then show how to use it

    1. Write the sequence in order. 2. Find each consecutive difference. 3. Compare the differences. 4. If they match, the sequence is an AP.

  3. 3

    Add one concrete example

    The sequence 1, 4, 7, 10 is an AP, but 1, 3, 6, 10 is not an AP because the differences are not equal.

  4. 4

    Avoid this incomplete answer

    A common wrong answer is to treat any increasing sequence as an AP without checking the actual gaps.

Definition

Choosing whether a sequence is an AP means checking if the differences between consecutive terms are all equal.

Example

The sequence 1, 4, 7, 10 is an AP, but 1, 3, 6, 10 is not an AP because the differences are not equal.

Rule to remember

Consecutive differences must be equal for an AP: a2 - a1 = a3 - a2 = a4 - a3 = ...

Memory hook

Equal gaps decide AP, not just a nice-looking pattern.

Examples and method

Worked example

For 2, 5, 9, 14, the differences are 3, 4, and 5. Since they are not equal, the sequence is not an AP.

Method to apply

1. Write the sequence in order. 2. Find each consecutive difference. 3. Compare the differences. 4. If they match, the sequence is an AP.

Diagram support

A difference table can show whether the pattern stays constant across the sequence.

How CBSE asks it

Questions may ask whether a given list is an AP, or they may ask students to justify the answer with differences.

Avoid common mistakes

Common confusion

Students often decide from the first two terms only and ignore the rest of the sequence.

Common wrong answer

A common wrong answer is to treat any increasing sequence as an AP without checking the actual gaps.

Exam tip

Check at least three consecutive differences before finalising your answer.

Quick check

What is the safest test to decide whether a sequence is an AP?

The safest test is to find the difference between each pair of consecutive terms. If all the differences are equal, the sequence is an AP.

Answer writing and exam use

1-mark answer

Choosing whether a sequence is an AP means checking if the differences between consecutive terms are all equal.

2-mark answer

Choosing whether a sequence is an AP means checking if the differences between consecutive terms are all equal. Consecutive differences must be equal for an AP: a2 - a1 = a3 - a2 = a4 - a3 = ... The sequence 1, 4, 7, 10 is an AP, but 1, 3, 6, 10 is not an AP because the differences are not equal.

3-mark answer

This skill is essential because students must not assume every regular-looking list is an AP. Some sequences increase unevenly, while others may follow a different pattern entirely. The safest method is to compare consecutive differences carefully and consistently. Consecutive differences must be equal for an AP: a2 - a1 = a3 - a2 = a4 - a3 = ... For 2, 5, 9, 14, the differences are 3, 4, and 5. Since they are not equal, the sequence is not an AP. Questions may ask whether a given list is an AP, or they may ask students to justify the answer with differences. A common wrong answer is to treat any increasing sequence as an AP without checking the actual gaps.
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