Arithmetic Progressions

What is an arithmetic progression?

An arithmetic progression (AP) is a list of numbers where the difference between any term and the one before it is always the same. That constant difference is called the common difference, $d$:

A sequence only counts as an AP if every consecutive pair shares that same difference, not just the first pair.

Worked example: does a situation form an AP?

Which of these situations produce an arithmetic progression?

• The taxi fare after each km, when the fare is ₹20 for the first km and rises by ₹8 for each additional km.
• The amount of air left in a cylinder, when a vacuum pump removes $\frac{1}{4}$ of the air remaining at each step.
• The cost of digging a well, when it costs ₹150 for the first metre and rises by ₹50 for each subsequent metre.
• The amount in an account each year, when ₹10,000 is deposited at 8% per annum compound interest.

Solution:

Taxi fare: ₹20, ₹28, ₹36, ₹44, …. Each term is 8 more than the last. The difference is constant ($d = 8$), so this is an AP.

Air in the cylinder: each step keeps $\frac{3}{4}$ of the previous amount, so the amounts shrink by a shared ratio, not a shared difference. The gaps between consecutive terms keep getting smaller. This is notan AP (it's a geometric progression instead).

Well-digging cost: ₹150, ₹200, ₹250, ₹300, …. A constant difference of $d = 50$, so this is an AP.

Compound interest:each year's amount is the previous amount times 1.08, so it's growing by a shared ratio again, not a shared amount. The yearly increase itself keeps growing. This is not an AP.

Worked example: three numbers in AP, given sum and product

Find three numbers in AP whose sum is 15 and whose product is 80.

Solution: Instead of calling the three numbers $a, a+d, a+2d$, it's easier to center them around the middle term: $a-d, \; a, \; a+d$. Their sum collapses nicely:

The product is a difference of squares, $(a-d)(a+d) = a^2-d^2$, times the middle term:

Either sign gives the same three numbers, just in reverse order: $2, \; 5, \; 8$.