How to Find the Common Difference of an AP

Quick answer: The common difference of an arithmetic progression is next term minus previous term. Check several consecutive differences: one matching pair alone does not establish an AP.

Worked example

Is 7, 3, −1, −5 an AP? Find its sixth term.

1. The differences are 3 − 7 = −4, −1 − 3 = −4, and −5 − (−1) = −4.
2. They agree, so a = 7 and d = −4.
3. a₆ = a + 5d = 7 − 20 = −13.

Check the result

Continuing the sequence gives −9 as the fifth term and −13 as the sixth.

Common mistake to avoid

Subtract in a consistent order. A decreasing AP has negative d. The nth term has n − 1 jumps from the first term.

Try it yourself

1. Find d for −2, 1, 4, 7. 2. Is 2, 4, 8, 16 an AP?

Answers and reasoning

1. d = 3. 2. No: the differences are 2, 4, 8, so they are not constant.

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