Subtract each term from the next

The idea

A list a₁, a₂, a₃, … is an AP if aₖ₊₁ – aₖ is the same for every k; then d = aₖ₊₁ – aₖ for any one k, always subtracting the kth term from the (k + 1)th.

Is it an AP?

Is the list 1, 1, 2, 3, 5, … an AP? Do not guess. Test it.

In an AP the number added is the same at every step. So we can look at the gap between each term and the next.

How would you test a list?

• Check that the gap between each term and the next is always the same
• Check that the list keeps getting bigger
• Check that the first and last terms are far apart

Test four lists

Tap the ? between two numbers to work out (next term) − (term before). Test every gap in each list.

The test

A list a₁, a₂, a₃, … is an AP if aₖ₊₁ − aₖ is the same for every k. Then d = aₖ₊₁ − aₖ. Always subtract the kth term from the (k + 1)th term, even when the (k + 1)th is smaller.

In 1, −1, −3, −5 the first difference is −1 − 1 = −2, not 1 − (−1) = 2. And once you know a list is an AP, one difference is enough to find d.

Notes

A list is an AP if aₖ₊₁ − aₖ is the same for every k, and then d = aₖ₊₁ − aₖ. Subtract the earlier term from the later one.

Check yourself

Does 2, 5, 8, 11, … form an AP?

Find d for the AP 8, 3, −2, −7, …

Answer: -5

3 − 8 = −5, −2 − 3 = −5, −7 − (−2) = −5. So d = −5.

Which list is not an AP?

The list 4, 10, 16, 22, … is an AP. What is the next term?

Answer: 28

d = 6, so the next term is 22 + 6 = 28.

• Yes. Every difference is 3. — correct. Yes! 5 − 2 = 3, 8 − 5 = 3, 11 − 8 = 3. The same every time.
• No. The terms are not all equal.. The terms of an AP do not have to be equal. The differences do.
• No. The list is not a pattern.. It is a pattern: each term is the one before plus 3.
• 2, 4, 8, 14, … — correct. Yes! The differences are 2, 4, 6. They are not all the same.
• −3, −1, 1, 3, …. The differences are 2, 2, 2. It is an AP.
• 50, 45, 40, 35, …. The differences are all −5. It is an AP.