Chapter 6.1 — Introduction to Arithmetic Progression

What Makes a Sequence a Progression

A progression is a sequence of numbers in which each term relates to the one before it through a consistent, rule-based pattern, rather than appearing at random. Each individual number appearing anywhere in the sequence, whatever its position, is called a term of the progression.

Every one of these counts as a progression — but they're not all built the same way. This chapter focuses on the simplest and most common kind first: sequences built by adding the same fixed number every time.

Two further sequences worth seeing here for contrast: 1, 2, 4, 7, 11, … increases by 1, then 2, then 3, then 4 — a genuine, describable rule, but not a fixed amount added each time, so it belongs to a different family of progressions entirely, one this chapter doesn't cover. Similarly, 2/3, 5/7, 8/11, 11/15, 14/19, … has a clear numerator and denominator pattern running through it, but neither the numerators alone nor the fraction values themselves change by any fixed amount from one term to the next. Both are entirely legitimate progressions under the broad definition above; neither one, however, is the specific kind — arithmetic — that the rest of this chapter is actually built around.

Arithmetic Progressions: A Fixed Step Every Time

An arithmetic progression (AP) is a sequence in which every single term, after the very first one, is obtained by adding or subtracting the same fixed number from the term that comes immediately before it. That fixed number is called the common difference of the progression, and it's conventionally written as the letter d.

In 3, 5, 7, 9, 11, …, each term is 2 more than the last: 3+2=5, 5+2=7, 7+2=9, 9+2=11. So this is an AP with common difference d = 2. Notice the phrase "except the first term" in the definition above — the very first term of an AP has no term before it to be compared against, so it's simply given as a starting value, not generated by the rule at all. Every term from the second one onward is what the common difference actually governs.

Four Common Differences, Four Different Patterns

The common difference doesn't have to be a small positive whole number at all — it can just as easily be negative, fractional, or decimal, and the sequence remains a genuine AP either way, without exception.

The second row is worth checking directly: 16 − 4 = 12, 12 − 4 = 8, 8 − 4 = 4 — each step subtracts 4, which is exactly the same as adding a common difference of −4. A "decreasing" AP is still an AP; the definition never required d to be positive. The sign of d alone tells you the overall direction a sequence moves: positive d means every term is larger than the one before, negative d means every term is smaller, and — as later exercises show — d = 0 is also perfectly valid, producing a sequence that never changes at all.

The General Form

Once you know just the first term a and the common difference d, every single other term of the AP follows automatically:

The second term adds d once, the third adds it twice, the fourth adds it three times — the coefficient sitting in front of d is always exactly one less than the term's position in the sequence. This pattern is the seed of the nth-term formula covered directly in the next exercise, where "one less than the position" becomes the (n−1) that multiplies d for any term aₙ. Equivalently, an AP can be defined by a single testable condition: for any AP a₁, a₂, a₃, …, aₙ,

This is genuinely the most useful way to think about checking whether a sequence is an AP at all: compute the difference between every pair of consecutive terms, and confirm every single one comes out identical. If even one pair disagrees, the sequence isn't an AP — no matter how regular the rest of it looks.

Naming Terms by Position

It helps to have a consistent way to refer to "the term in position n" without writing out the whole sequence every time. The standard notation writes a₁ for the first term, a₂ for the second, and generally aₙ for the nth term — so an AP with first term a and common difference d can be written a₁, a₂, a₃, …, aₙ, with a₁ = a. When a progression has a fixed number of terms rather than continuing forever, the very last term is often given its own letter, l, purely as a convenience for formulas needing to refer to "wherever the sequence ends" without knowing n in advance.

This notation isn't just bookkeeping — it's what makes it genuinely possible to state a fact about "any AP whatsoever" in a single line, rather than repeating essentially the same argument separately for every specific numerical example encountered. The equal-differences condition above, a₂−a₁=a₃−a₂=…=aₙ−aₙ₋₁=d, is a claim about every arithmetic progression at once, precisely because it's phrased in terms of positions rather than particular numbers.

Where This Chapter Goes Next

This introduction covers only arithmetic progressions and the vocabulary — term, common difference, position notation — that the rest of the chapter is built on, roughly the chapter's first half. Exercise 6.1 applies the consecutive-difference test to real situations and abstract sequences alike; Exercise 6.2 turns the general form above into a direct formula for any term; Exercise 6.3 derives a genuinely fast way to add up an entire AP at once; and Exercise 6.4 introduces a second kind of progression entirely — one built by multiplying by a fixed ratio instead of adding a fixed difference, with Exercise 6.5 giving it the same nth-term treatment Exercise 6.2 gives arithmetic progressions. For the equation-solving skills this chapter assumes throughout, see Chapter 5's introduction, which several later word problems in this chapter lean on directly.