medium difficulty6 min

Common difference

The common difference is the fixed number added to each term of an AP to get the next term.

In one line: d means the same step repeats every time.

Practice This Concept

Understand it

If you subtract any term from the next term in an AP, you get the same value every time.
That value is the common difference, written as d.
It can be positive, negative, or zero.
This is one of the most important ideas in the chapter because almost every AP formula uses d.

Example

In 15, 11, 7, 3, ... the common difference is -4 because each term is 4 less than the previous term.

Formula and rule

For an AP, d = a2 - a1 = a3 - a2 = a4 - a3 and so on.

See it worked out

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

Worked example

1For 3, 8, 13, 18, ... we get d = 8 - 3 = 5, and checking again gives 13 - 8 = 5.
2So the common difference is 5.

Method you can reuse

1Take two consecutive terms.
2Subtract the first from the next term.
3Check another pair to confirm.
4Keep the sign as it appears from the sequence order.

Check yourself

Answer the question in your head before opening the answer.

Try it yourself

What does the common difference tell you in an AP?

Think first, then show answer

The common difference tells you the fixed number added to or subtracted from each term to get the next term. It shows the regular step size of the sequence.

Common wrong answer

A common wrong answer is to use the absolute difference without sign. That hides whether the AP is increasing or decreasing.

Don't mix this up

Students often write the difference as a larger number minus a smaller number in the wrong order, which changes the sign.

Exam tip

Keep the order of subtraction the same as the order of the sequence: next term minus previous term.

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 students to find d from a sequence, identify a mistake in sign, or use d in a formula.

1
Enough for 1 mark
The common difference is the fixed number added to each term of an AP to get the next term.
2
Add for 2 marks
Example: In 15, 11, 7, 3, ... the common difference is -4 because each term is 4 less than the previous term.
3
Add for 3 marks
That value is the common difference, written as d.

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