How big is each step?

The idea

The fixed number is the common difference d of the AP, and it can be positive, negative or zero; for terms a₁, a₂, …, aₙ, a₂ – a₁ = a₃ – a₂ = … = aₙ – aₙ₋₁ = d.

Three lists

Look at three lists: 1, 2, 3, 4, … and 100, 70, 40, 10, … and 3, 3, 3, 3, …

In each one, the next term is the term before it plus a fixed number. In the first list it is +1.

In the list 100, 70, 40, 10, … what kind of number is added to each term?

• A positive number
• A negative number
• Nothing can be added, because the list goes down

Turn the step dial

The bars are the terms of an AP. The dial d is the fixed number added each time. Turn it and watch the bars.

Common difference

The fixed number is called the common difference, written d. For terms a₁, a₂, a₃, …: d = a₂ − a₁ = a₃ − a₂ = … = aₙ − aₙ₋₁.

d can be positive (the bars climb), negative (the bars fall) or zero (all the terms are equal, like 3, 3, 3, 3, …). To find d, subtract a term from the term after it.

Notes

d = (any term) − (the term before it). It can be positive (terms climb), negative (terms fall) or zero (terms stay the same).

Check yourself

Find d for the AP 10, 13, 16, 19, …

Answer: 3

13 − 10 = 3, 16 − 13 = 3, 19 − 16 = 3. So d = 3.

Find d for the AP 20, 15, 10, 5, …

Answer: -5

15 − 20 = −5, 10 − 15 = −5, 5 − 10 = −5. So d = −5.

In the AP 7, 7, 7, 7, … what is d?

Find d for the AP −1.0, −1.5, −2.0, −2.5, …

Answer: -0.5

−1.5 − (−1.0) = −0.5, and −2.0 − (−1.5) = −0.5. So d = −0.5.

• 0 — correct. Yes! Adding 0 gives the same term again. d can be zero.
• 7. 7 is the term, not the step. The next term minus the term before it is 7 − 7 = 0.
• There is no d. A fixed number is added each time: the number 0. So it is an AP, with d = 0.