Two numbers are enough

The idea

To know an AP, both its first term a and its common difference d are needed; once a and d are known, the AP can be listed.

Is the first term enough?

An AP starts at 6. Can you write the whole list?

Here are three lists that all start at 6: 6, 9, 12, … and 6, 3, 0, … and 6, 6, 6, …

What else do you need, to pick exactly one of these lists?

• The common difference d
• The last term
• Nothing. The first term 6 is enough.

Make the target lists

Turn the dial a (the first term) and the dial d (the step). Make each target list.

Both a and d

To know an AP you need both the first term a and the common difference d. Once you know a and d, you can list the AP: a, a + d, a + 2d, …

The same a with a different d gives a different list (6, 9, 12, … and 6, 3, 0, …). The same d with a different a gives a different list too. With a = 2 and d = 0 the list is 2, 2, 2, 2, …

Notes

An AP is fixed by its first term a and its common difference d. Know both, and you can write the list.

Check yourself

An AP has a = 5 and d = 4. What is its 3rd term?

Answer: 13

5, 9, 13. The 3rd term is 13.

Which pair of numbers fixes the AP 10, 7, 4, 1, …?

An AP has a = −7 and d = −2. What is its 3rd term?

Answer: -11

−7, −9, −11. The 3rd term is −11.

Two APs both start at 8, but their lists are different. For example 8, 10, 12, … and 8, 5, 2, … What must be different?

• a = 10 and d = −3 — correct. Yes! The list starts at 10, and 7 − 10 = −3.
• a = 10 and d = 3. d is (next term) − (term before it) = 7 − 10 = −3. The list goes down.
• a = 7 and d = −3. a is the first term of the list, which is 10.
• Their common differences — correct. Yes! The first term is the same (8), so the lists can differ only in d.
• Their first terms. Both start at 8, so a is the same.
• Nothing. The first term is enough to fix the list.. The lists are different, so a is not enough. You need d as well.