Pearson Edexcel International GCSE in Mathematics (Specification A) · 4MA1 · Free · no account needed

Linear Sequences & the nth Term

In one sentence. Finding the rule that generates a sequence — and using it to predict, test and bound terms without listing hundreds of them.

Sequences — one of 2 modules in this strand, and one of 53 in Pearson Edexcel International GCSE Mathematics. Both tiers.

This module ends with Carl Friedrich Gauss. Learn Linear Sequences & the nth Term, teach it, then defend it against Carl Friedrich Gauss.

Where this sits in the specification

Specification 4MA1, 3.1 Sequences. Incandio is aligned to this specification; it is not published by, endorsed by or affiliated with Pearson, and the sub-topic numbers are given so you can check every module against your own copy.

  • 3.1 — Sequences

1 · Understand it

What it is

A linear (arithmetic) sequence climbs by the same amount each step: 5, 9, 13, 17 climbs by 4. Its nth term is a formula giving ANY term directly: here 4n + 1, so the 50th term is 4(50) + 1 = 201 without writing out 49 others.

Why it works

The common difference IS the multiplier of n: climbing 4 each step means the sequence shadows the 4-times table. Comparing with that table (4, 8, 12, 16 against 5, 9, 13, 17) shows the whole sequence runs exactly 1 higher — hence 4n + 1. The formula is 'times-table plus correction'.

When a question is asking for it

'Find the nth term' directly; 'is 100 a term?' (set the formula equal to 100 — a whole-number n means yes); 'the first term greater/less than…' (solve an inequality); and pattern-growth questions with matchsticks or tiles, which are sequences wearing pictures.

How to hold it in your head

nth term = (difference) × n + (what the 'zeroth term' would be). Step BACK once from the first term: 5 − 4 = 1, and that back-step is the correction: 4n + 1. One subtraction, no guessing.

2 · The mistakes that cost marks

  • Answering 'add 4' for the nth term. That's the term-to-term rule — the question wants a formula in n.
  • Using the first term as the correction: 5, 9, 13 → 4n + 5 ✗. Test it: 4(1) + 5 = 9, wrong first term. The correction is first term MINUS difference.
  • Deciding '100 is in the sequence because it's even'. Test properly: solve 3n + 4 = 100 and check n is a whole number.
  • For decreasing sequences, forgetting the difference is negative: 20, 17, 14 has difference −3, giving 23 − 3n.

3 · Worked examples

Laid out the way an examiner wants working shown — every step is a mark.

Find the nth term of 5, 9, 13, 17, …

  1. Common difference: +4, so the rule contains 4n.
  2. Step back from the first term: 5 − 4 = 1.
  3. Rule: 4n + 1. Test: 4(2) + 1 = 9 ✓

Answer: 4n + 1

The nth term of a sequence is 5n − 3. Is 150 a term of the sequence?

  1. Set 5n − 3 = 150 → 5n = 153 → n = 30.6.
  2. n is not a whole number — there is no '30.6th term' — so 150 is NOT in the sequence.
  3. (Check the neighbours: term 30 is 147 and term 31 is 152, so the sequence jumps straight over 150. A whole-number n would have meant yes.)

Answer: No — n = 30.6 is not a whole number

4 · Prove it — the five questions

The five questions climb Grade 6 → 9+ and are marked inside Incandio on your own device, against authored accepted forms, with a diagnosis of the wrong answer you actually gave. The answers stay in the app so that the practice is worth doing; the questions themselves are here.

  1. Grade 6 — Find the nth term of the sequence 5, 9, 13, 17, … (Type it like 4n+1.)
  2. Grade 7 — Find the nth term of the sequence 3, 10, 17, 24, … (Type it like 7n-4.)
  3. Grade 8 — The nth term of a sequence is 3n + 4. Work out the largest term of the sequence that is less than 100.
  4. Grade 9 — A sequence has nth term 7n − 5. Which term of the sequence is equal to 156?
  5. 9+ — A sequence has nth term 23 − 3n. Work out the position of the first NEGATIVE term.

Ask the person who made it

Inside Incandio you can put a question to Ada Lovelace, who answers knowing exactly which module you have just finished: “Your notes on the Analytical Engine were all about general rules that generate endless results — like an nth term generating a whole sequence. Why is capturing the RULE so much more powerful than listing the numbers?”

Think you understand it?

Reading is the start. On Incandio a topic counts as mastered only once you have taught it and defended it — the two things a revision guide cannot ask of you.

Incandio this topic → Prove it. Teach it. Defend it. What does “Incandio it” mean?

  1. Prove it — five questions on Linear Sequences & the nth Term, marked on the spot, that tell you what was missing rather than just what was wrong.
  2. Teach it — Explain how Linear Sequences & the nth Term works — and why — to Ember, an AI pupil who does not have it yet. The method matters more than the vocabulary.
  3. Defend it — Argue Linear Sequences & the nth Term against Carl Friedrich Gauss — three rounds, marked against descriptors you can read before you begin.
  4. Then the exam — read all 11 authored exam-style questions on this unit, each with its command word and its tariff, then have them marked on Incandio with the mistake you actually made named.

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