IGCSE Maths Sequences notes

A sequence is an ordered list; each number is a term. The term-to-term rule says how the list moves: 7, 10, 13, 16 adds 3 each time, so the next terms are 19 and 22. Some rules mix operations: 3, 7, 15, 31 is “×2 then +1” — check it against every term, not just the last: 3×2+1 = 7 ✓, 7×2+1 = 15 ✓, 15×2+1 = 31 ✓, so the next term is 31×2+1 = 63.

Term-to-term rules answer “what is the next term?” — but they are hopeless for “what is the 50th term?”. Adding 3 fifty times is slow and error-prone. That is why the syllabus wants the nth term: a formula with the position number n in it, so any term can be computed in one line.

If a sequence changes by the same amount every term, it is linear and its nth term looks like dn + c. The common difference d is the coefficient of n: 5, 8, 11, 14 grows by 3, so it starts as 3n + …. To find the constant, put n = 1: 3(1) + c = 5, so c = 2 and the nth term is 3n + 2. Check with n = 2: 3(2) + 2 = 8 ✓.

Decreasing sequences give negative coefficients: 10, 7, 4, 1 falls by 3, so −3n + c; with n = 1, −3 + c = 10, c = 13 → 13 − 3n. The nth term answers position questions instantly: the 20th term of 6n − 1 is 6(20) − 1 = 119; and to find which term equals 57 in 4n − 3, solve 4n − 3 = 57 → n = 15, so 57 is the 15th term.

When the differences change by a fixed amount, the sequence is quadratic. Take 2, 5, 10, 17, 26: differences 3, 5, 7, 9; second differences 2, 2, 2. Half the second difference is the coefficient of n²: try n². Subtracting n² from the terms leaves 1, 1, 1, 1, 1, so the nth term is n² + 1. Check n = 4: 16 + 1 = 17 ✓.

The same recipe works when leftovers are not constant: 3, 8, 15, 24 has differences 5, 7, 9 (second difference 2), so start with n²; subtracting gives 2, 4, 6, 8 — that is 2n — so the nth term is n² + 2n. Cubic sequences show constant third differences: 1, 8, 27, 64 is simply n³; 2, 16, 54, 128 is 2n³ (each term is 2 × 1, 2 × 8, 2 × 27, 2 × 64). Spotting “is each term k times a cube?” is usually faster than grinding third differences.

Extended. When each term is multiplied by the same number, the sequence is geometric. Its nth term is a × rⁿ⁻¹, where a is the first term and r the multiplier: 5, 10, 20, 40 multiplies by 2, so the nth term is 5 × 2ⁿ⁻¹ (n = 1 gives 5 × 1 = 5 ✓). Halving works the same way: 80, 40, 20, 10 multiplies by ½, giving 80 × (½)ⁿ⁻¹.

Growth questions ask when a sequence passes a threshold: for 3, 6, 12, 24, … (nth term 3 × 2ⁿ⁻¹), the first term above 1000 is the 10th, since 3 × 2⁸ = 768 (too small) and 3 × 2⁹ = 1536. Decimal ratios are fair game too: 625, 125, 25, 5 divides by five each time — nth term 625 × (1⁄5)ⁿ⁻¹ — and asking for the term equal to 1⁄25 means solving 625 × (1⁄5)ⁿ⁻¹ = 1⁄25, i.e. the 7th term.