Where progressions show up in real life

It is easy to meet arithmetic and geometric progressions as a list of numbers on a page, $3, 7, 11, 15, \ldots$ or $2, 6, 18, 54, \ldots$, and treat them as a purely symbolic exercise. But almost every progression question in Add Math is describing something that changes over time, or over a sequence of steps, in one of exactly two ways: by adding the same amount each time, or by multiplying by the same amount each time.

That distinction is the whole chapter, really. An arithmetic progression (AP) adds a fixed amount, the common difference $d$, at every step.

A geometric progression (GP) multiplies by a fixed factor, the common ratio $r$, at every step. Almost any real situation that repeats in a regular way falls into one of these two patterns, and recognising which one is usually the first and most important decision in a word problem.

• Adds the same amount each step → arithmetic progression.
• Multiplies by the same factor each step → geometric progression.
• Neither, the amount added or the factor itself keeps changing → not a simple progression at all.

An arithmetic progression shows up whenever a quantity increases, or decreases, by a fixed, flat amount at every step, with no compounding involved. A worker whose monthly pay rises by a fixed sum each year, seats arranged in rows where each row has a fixed number more seats than the row before, or a savings plan where the same fixed amount is deposited every month, all follow this pattern.

The two formulas that describe the pattern are worth being able to reach for without hesitation:

$$T_{n} = a + (n-1)d$$

$$S_{n} = \frac{n}{2}\left[2a+(n-1)d\right]$$

$T_{n}$ is the value at step $n$, the $n$th term, while $S_{n}$ is the running total of the first $n$ terms. If a hall has 20 seats in the front row and each row behind it has 2 more seats than the row in front, the number of seats in row 12 is $T_{12}=20+(12-1)(2)=42$, and the total number of seats across the first 12 rows is $S_{12}=\frac{12}{2}\left[2(20)+(11)(2)\right]=372$.

A geometric progression shows up whenever a quantity grows, or shrinks, by a fixed percentage or fixed factor at every step, which is what makes it feel faster than arithmetic growth over time. Compound interest on savings, the population of bacteria that doubles every hour, the value of a car that depreciates by a fixed percentage each year, and a ball that bounces to a fixed fraction of its previous height each time, all of these are geometric, not arithmetic, because the change scales with the current amount rather than staying flat.

The matching formulas are:

$$T_{n} = ar^{n-1}$$

$$S_{n} = \frac{a(1-r^{n})}{1-r}, \quad r \ne 1$$

If RM1,000 is invested and grows by 5% each year, the amount after each year forms a geometric progression with $a=1000$ and $r=1.05$; after 4 years the value is $T_{5}=1000(1.05)^{4}\approx 1215.51$ (the fifth term, since the first term is the starting amount before any growth has been applied). Notice that $r$ here is 1.05, not 0.05, a common slip is writing the growth rate itself as $r$ instead of $1+$rate.

One of the more surprising ideas in this chapter is that adding infinitely many terms can still give a finite, sensible answer, but only for a geometric progression, and only when the common ratio satisfies $-1<r<1$. In that case each new term is smaller than the one before it, by the same factor every time, and the running total settles toward a fixed value rather than growing without bound.

$$S_{\infty} = \frac{a}{1-r}, \quad -1<r<1$$

A bouncing ball is the classic picture for this: drop a ball from 2 metres, and suppose it always bounces back up to $\frac{3}{4}$ of the height it fell from. The heights of the bounces after the first drop form a geometric progression with $a=2\left(\frac{3}{4}\right)=1.5$ and $r=\frac{3}{4}$.

The total distance the ball would travel bouncing, if it bounced forever, is finite: $S_{\infty}=\frac{1.5}{1-0.75}=6$ metres, even though the ball, in principle, bounces an infinite number of times.

The same idea explains why a recurring decimal such as $0.\overline{3}$ equals exactly $\frac{1}{3}$: written as $0.3+0.03+0.003+\cdots$, it is a geometric progression with $a=0.3$ and $r=0.1$, and $S_{\infty}=\frac{0.3}{1-0.1}=\frac{1}{3}$.

The mathematics in this chapter is rarely the hard part, the formulas are short, and calculators handle the arithmetic. What actually separates a confident answer from a stuck one is reading the situation correctly: is the quantity being added to, or multiplied?

What exactly is the first term, and does it correspond to $n=1$ or does the counting start somewhere else in the story?

• Underline the phrase that tells you the pattern, "increases by", "the same amount more", and "more than the previous" usually signal arithmetic; "increases by a percentage", "grows by a factor", and "each time as much again" usually signal geometric.
• Check what counts as the first term. If a savings account already has RM500 before the first deposit is added, that RM500 might not be $a$, read carefully whether the progression starts before or after the first change.
• Decide whether the question wants a specific term ($T_{n}$), a running total ($S_{n}$), or the long-run total ($S_{\infty}$), these ask genuinely different questions, and the wrong formula gives a wrong answer even with correct $a$, $d$, or $r$.

The formulas for progressions are short enough to memorise in an afternoon; what takes longer to build is the instinct for reading a paragraph of words and immediately seeing "arithmetic" or "geometric" inside it. That instinct comes from working through enough real situations, out loud, with someone who can point out exactly which phrase gave the pattern away, and exactly where a growth rate got mistaken for $r$ itself.

Our teachers are experienced; lessons run online and are taught in English, while SPM papers are set bilingually in Bahasa Melayu and English.

If progressions, particularly the word problems built around them, feel harder than the algebra suggests they should, a one-hour paid trial class at the teacher's own rate (from RM50 per hour, depending on experience) is a fair way to see whether that kind of one-to-one attention helps. We will not promise a particular grade, but we can help the pattern-spotting side of this chapter become second nature.