Word Problems

1. Introduction: Turning a Story Into a List of Numbers

You already know the two big tools of this chapter. An AP — Arithmetic Progression — is a list of numbers where each term jumps by the same fixed amount. That fixed jump is the common difference, written $d$. The first term is written $a$.

You also know two formulas:

• The $n$th term: $a_n = a + (n-1)d$ — this finds one term, the value at position $n$.
• The sum of the first $n$ terms: $S_n = \dfrac{n}{2}\big[2a + (n-1)d\big]$ — this finds the total of the first $n$ terms.

So what is left to learn? This. In a real question, nobody hands you "$a = 200$, $d = 50$, find $S_{30}$." Instead they hand you a story: a worker whose salary rises every year, a stack of logs that narrows as it goes up, a woman who saves a little more each month. Your job is to read the story, spot the hidden AP, and decide which of the two tools to reach for.

Stop scrolling. Try it in your head before reading on. A shopkeeper sells 5 umbrellas on Monday, 8 on Tuesday, 11 on Wednesday — three more each day. Is that an AP? What is $a$, and what is $d$?

That is the whole skill of this lesson: read the words, find $a$ and $d$, decide between "one term" and "the total", then compute. By the end, a word problem will not scare you — you will calmly turn it into a few lines of working.

You can now say what this lesson is about: solving real-life problems by spotting the hidden AP inside the words.