medium difficulty5 min

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.

In one line: Equal gaps decide AP, not just a nice-looking pattern.

Understand it

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.

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.

Formula and rule

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

See it worked out

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

Worked example

1For 2, 5, 9, 14, the differences are 3, 4, and 5.
2Since they are not equal, the sequence is not an AP.

Method you can reuse

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

Visual tip

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

Check yourself

Answer the question in your head before opening the answer.

Try it yourself

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

Think first, then show answer

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.

Common wrong answer

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

Don't mix this up

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

Exam tip

Check at least three consecutive differences before finalising your answer.

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

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

1
Enough for 1 mark
Choosing whether a sequence is an AP means checking if the differences between consecutive terms are all equal.
2
Add for 2 marks
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.
3
Add for 3 marks
Some sequences increase unevenly, while others may follow a different pattern entirely.

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