Chapter 5: Arithmetic Progressions

Take a number. Keep adding the same fixed amount. You get an arithmetic progression , one of the simplest and most useful patterns in mathematics. The salaries on an annual-increment scale, the rows of seats in an auditorium, the heights of stacked boxes, the lengths of pendulum swings (approximately) all follow this pattern.

This chapter introduces sequences , ordered lists of numbers obeying a rule , and focuses on the special case where the rule is "add the same number $d$ each time". Two questions become central: what is the $n$th term, and what is the sum of the first $n$ terms? Both have crisp closed-form formulas.

For boards, expect $1$-mark MCQs on whether a list is an AP, $2$-mark questions on the common difference and the $n$th term, $3$-mark problems on finding terms given some conditions or summing a finite AP, and $4$- to $5$-mark applications: salary, seats, brick stacking, debt schedules, and so on. The arithmetic is light; the cleverness is in setting up.

Beyond board-exam utility, APs are the simplest stepping stone into infinite series and calculus. Whenever you see "regular spacing", an AP is hiding.

What's inside

• Sequences and APs , definition, common difference, simple examples.
• General ($n$th) term of an AP , the formula $a_n = a + (n - 1) d$.
• Sum of the first $n$ terms , derivation and the two equivalent formulas.
• Word problems and applications , salary, seating, stacking, debts.
• Arithmetic mean and consecutive APs , small but exam-useful results.

Key results / Formula card

For an AP with first term $a$, common difference $d$, and $n$ terms:

Always identify $a$ and $d$ first. Almost every AP problem reduces to two simultaneous equations in $a$ and $d$.