Arithmetic Progressions

An arithmetic progression adds a constant difference $d$ each step: $a$, $a + d$, $a + 2d$, … Test for it by subtracting consecutive terms, constant difference, AP confirmed.

The formulas

• nth term: $u_n = a + (n - 1)d$
• Sum of n terms: $S_n = \tfrac{n}{2}\left[2a + (n - 1)d\right]$ or $S_n = \tfrac{n}{2}(\text{first} + \text{last})$

All three memorised. The second sum form is criminally underused: when you know the last term (or can get it cheaply), $\tfrac{n}{2}(\text{first} + \text{last})$ halves the algebra.

The standard question: translate facts into equations

The 4th term of an AP is 14 and the sum of the first 10 terms is 185. Find $a$ and $d$. $u_4 = a + 3d = 14$ ← translation line 1 (M) $S_{10} = 5(2a + 9d) = 185 \to 2a + 9d = 37$ ← translation line 2 (M) Solve simultaneously: from the first, $a = 14 - 3d \to 28 - 6d + 9d = 37 \to 3d = 9 \to$ $d = 3$, $a = 5$

Each given fact becomes exactly one equation, write the translations before solving, because they carry the method marks even if the algebra slips. The classic trap is the off-by-one: the 4th term uses $3d$, the 10th uses $9d$. Saying "$(n - 1)$" aloud as you substitute kills the error.

Other regulars: “which term equals 95?” → set $u_n = 95$, solve for $n$ (must come out a positive integer, if not, re-check); “how many terms are needed for the sum to exceed 1000?” → $S_n > 1000$, a quadratic inequality in $n$, answered with the integer $n$ that first qualifies.

Word-problem costumes

APs arrive dressed as salaries rising by a fixed annual increment, seats per row increasing by a constant, stacked logs. The detection question: is a constant amount added each step? (Constant ratio means a GP.) Define $a$ and $d$ explicitly from the story (”$a$ = first-year salary = 30 000, $d$ = 2 000”) before touching formulas, examiners credit the setup.

Common mistakes

• $(n - 1)$ slips: $u_{10}$ computed with $10d$
• Sum and term formulas swapped
• $n$ from “which term” equations left as a fraction without alarm
• AP formulas applied to a sequence never tested for constant difference
• Word problems started without defining $a$, $d$, $n$ in the story’s terms

Full topic context: Series notes · the multiplicative twin: geometric progressions.