Arithmetic Progressions

An arithmetic progression (AP) is a sequence of numbers in which the difference between any two consecutive terms is always the same. That fixed gap is called the common difference, written d. For example, 3, 7, 11, 15, ... is an AP because you add 4 each time, so d = 4. The first term is called a. If d is positive the AP increases; if d is negative it decreases (like 10, 7, 4, 1, ...). Everyday lists are often APs: a fixed monthly saving, seats increasing row by row in a hall, or a taxi fare that rises by a fixed amount per kilometre.

Instead of writing out every term to reach, say, the 50th, we use a formula. The first term is a, the second is a + d, the third is a + 2d, and so on, so the nth term is an = a + (n - 1)d. Notice the multiplier of d is always one less than the term number, because the first term has had no jumps added yet. This single formula lets you jump straight to any term, check whether a number belongs to an AP, or find how many terms an AP has.

To add the first n terms we use the story of young Gauss, who added 1 to 100 in seconds. Pair the first term with the last, the second with the second-last, and so on: every pair has the same total, a + l, where l is the last term. With n terms there are n such pairings counted twice, giving Sn = n/2 (a + l). Since the last term l = a + (n - 1)d, we can also write Sn = n/2 [2a + (n - 1)d], which works even when the last term is unknown.

APs model any situation with steady, equal change. Salaries that rise by a fixed annual increment, loan instalments, the number of logs stacked in tapering rows, production that grows by a constant amount each year, and simple interest (which grows by the same amount yearly) are all APs. The two formulas, the nth term and the sum, are enough to solve almost every Class 10 board problem on this topic, so practising how to read a word problem and pick out a, d and n is the real skill.