GCSE Maths Sequences & Nth Term Revision

Overview

What you’ll learn

Generate terms from a term-to-term rule and from a position-to-term (nth term) rule.
Find the nth term of a linear sequence and use it to find any term or test membership.
Recognise triangular, square and cube number sequences on sight.
Continue Fibonacci-type sequences and geometric progressions, including surd ratios (Higher).
Find the nth term of a quadratic sequence using second differences (Higher).
Describe sequence rules precisely in words and algebra (Higher).

Specification checklist (AQA 8300 · Edexcel 1MA1 · OCR J560)

Statements are paraphrased from the shared DfE subject content and grouped by tier. Foundation content is studied by every student; Higher statements are examined on Higher-tier papers only. QuizQuip is not affiliated with AQA, Pearson Edexcel or OCR.

Foundation — every student

10.1Generate terms of a sequence from either a term-to-term rule or a position-to-term rule.
10.2Recognise and use arithmetic (linear) sequences, deduce expressions for their nth term, and use the nth term to calculate terms and decide whether a number belongs to the sequence.
10.3Recognise and use the sequences of triangular, square and cube numbers, and simple alternating or growing patterns built from them.

Higher — Higher tier only

10.4Recognise Fibonacci-type sequences and simple geometric progressions (rⁿ where n is a position and r is rational and positive, or a surd), and deduce nth terms of quadratic sequences. (Higher)
10.5Continue and describe more complex patterns, explaining rules in words and with algebra. (Higher)

Key points

1Term-to-term rules step from one term to the next (add 3); position-to-term rules jump straight to the nth term (3n + 1).
2Linear sequence nth term: nth = dn + (first term − d), where d is the common difference.
3Is 100 a term of 4n − 1? Solve 4n − 1 = 100: n = 25.25 — not a whole number, so no.
4Triangular 1, 3, 6, 10, 15; squares 1, 4, 9, 16, 25; cubes 1, 8, 27, 64, 125.
5Fibonacci-type sequences add the two previous terms: 1, 3, 4, 7, 11, 18 (Higher).
6Geometric progressions multiply by a fixed ratio: 2, 6, 18, 54 (×3); surd ratios stay exact, like 2, 2√3, 6 (Higher).
7Quadratic sequences have constant second differences; nth = (second diff ÷ 2)n² + bn + c (Higher).
8The nth term is a position-to-term machine: feed in n, get the term — no stepping required.

Notes · 1 of 4

Generating terms

Two kinds of rule, two ways of building the list.

A term-to-term rule tells you how to step: “add 3.5, starting at 2” builds 2, 5.5, 9, 12.5, 16. A position-to-term rule gives the term straight from its position: the rule “double the position and subtract 1” builds 1, 3, 5, 7 — each term is 2n − 1. Position-to-term rules are more powerful: the 100th term needs no 99 intermediate steps.

Describing a sequence well says both the rule and the start: “starts at 40 and halves” gives 40, 20, 10, 5, 2.5. Vague descriptions (“it goes down”) earn no marks — the rule must let a stranger reproduce the list exactly.

Notes · 2 of 4

The nth term of a linear sequence

The common difference is the coefficient of n; the constant fixes the start.

An arithmetic (linear) sequence steps by the same amount every time — that step is the common difference, and it becomes the coefficient of n. For 5, 8, 11, 14 the difference is 3, so the nth term starts 3n; the first term 5 is 2 more than 3 × 1, so the nth term is 3n + 2. In general: nth = dn + (first − d).

The nth term answers two exam questions. Find a term: the 20th term of 4n − 1 is 4 × 20 − 1 = 79. Test membership: is 100 a term? Solve 4n − 1 = 100, giving n = 25.25 — not a whole number, so 100 is not in the sequence. A fractional or negative n always means “no”, and showing that solving step is the method mark.

Notes · 3 of 4

Special sequences

Triangular, square, cube — the sequences everyone recognises.

The exam loves three families. Triangular numbers 1, 3, 6, 10, 15 grow by 1, 2, 3, 4… (dots stack into triangles). Square numbers 1, 4, 9, 16, 25 are n². Cube numbers 1, 8, 27, 64, 125 are n³. Spotting these instantly saves time — a question may dress them in context (“stacking cannonballs”) but the pattern is the pattern.

Alternating and combined patterns appear too: 100, 97, 94, 91 (subtract 3) or 1, 2, 4, 8, 16 (double). For pattern-built sequences, describe the rule exactly — “add 4, then add 5, then add 6” is a valid term-to-term description even when no simple nth term is expected at Foundation.

Notes · 4 of 4

Fibonacci-type and geometric sequences

Add the last two — or multiply by a fixed ratio. (Higher)

Higher. A Fibonacci-type sequence adds the two previous terms: 1, 3, 4, 7, 11, 18 (1 + 3 = 4, 3 + 4 = 7, …). With algebra, a rule like “each term is the sum of the previous two, starting a and b” builds a, b, a + b, a + 2b + … — exam questions ask for the next term or an expression for one of them.

A geometric progression multiplies by a fixed ratio each step: 2, 6, 18, 54 (×3); 8, 4, 2, 1, ½ (×½); and surd ratios stay exact — 2, 2√3, 6, 6√3 multiplies by √3, since (√3)² = 3 makes every second term rational. Quadratic sequences have constant second differences: 5, 8, 13, 20 steps +3, +5, +7, and those steps grow by 2 each time. Halve the second difference for the n² coefficient: here n² + 4 (check: 9 + 4 = 13 ✓).

Revise

Key terms

The words examiners expect you to use precisely.

Term: One number in a sequence.The 3rd term of 5, 8, 11 is 11.
Term-to-term rule: A rule stepping from each term to the next.“Add 4”: 2, 6, 10, 14.
Position-to-term rule: A rule giving each term from its position, i.e. the nth term.4n − 1 gives 3, 7, 11, 15.
Common difference: The constant step of an arithmetic sequence.5, 8, 11 steps by 3.
Arithmetic sequence: A sequence with a constant difference — the nth term is linear.3n + 2.
Geometric progression: A sequence multiplied by a fixed ratio each step.2, 6, 18, 54 (×3).

Show 2 more key terms

Second difference: The difference of the differences, constant for quadratic sequences.5, 8, 13, 20 steps +3, +5, +7.
Triangular numbers: 1, 3, 6, 10, 15 — growing by 1, 2, 3, 4.Dots stacked as triangles.

Revise

Common mistakes

Where students most often lose marks in this chapter — and what to do instead.

Mistake: Reading 4n − 1 and computing the first term as 4.
The first term uses n = 1, so 4 × 1 − 1 = 3.
Instead: Substitute n = 1, 2, 3 to list terms.
Mistake: Forgetting the constant: 5, 8, 11 has nth term 3n.
3n starts at 3, not 5.
Instead: nth = 3n + 2 (first term minus difference).

Show 10 more mistakes

Mistake: Declaring 100 a term because 4n − 1 ≈ 100.
Membership needs an exact whole-number position.
Instead: 4n − 1 = 100 gives n = 25.25, so 100 is not a term.
Mistake: Confusing term-to-term with position-to-term rules.
“Add 3” does not give the 50th term without 49 steps.
Instead: The nth term is the efficient position-to-term description.
Mistake: Halving the first difference for a quadratic sequence’s n² coefficient.
It is the second difference that halves.
Instead: Second difference 2 → n² coefficient 1; second difference 6 → 3n².
Mistake: Calling 2, 6, 18, 54 an arithmetic sequence.
The steps 4, 12, 36 are not constant; the ratios are.
Instead: It is geometric with ratio 3.
Mistake: Continuing a Fibonacci-type sequence by adding a fixed number.
Each term adds the previous term, not a constant.
Instead: 1, 3, 4, 7, 11: each term is the sum of the two before it.
Mistake: Missing the alternating pattern: 2, 5, 3, 6, 4, 7 continued as 8.
The steps alternate +3 and −2, so the next step is −2, not +3.
Instead: Write the steps out (+3, −2, +3, −2, +3); the next term is 5.
Mistake: Using n = 0 to check an nth term.
GCSE positions run from n = 1.
Instead: Test n = 1 and n = 2 against the given terms.
Mistake: Subtracting the wrong n² multiple: using n² when the second difference is 4.
Second difference 4 means the n² coefficient is 2.
Instead: Second difference ÷ 2 is the n² coefficient.
Mistake: Leaving a surd ratio sequence as rounded decimals.
Exactness is the point of a surd progression.
Instead: 2, 2√3, 6, 6√3 with ratio √3.
Mistake: Describing a rule as “it goes up”.
Vague rules cannot reproduce the sequence.
Instead: “Add 3 each time, starting at 5” or “nth = 3n + 2”.

Check yourself

Quick quiz

30 questions with instant explanations. About 5 minutes.

Question 1 of 3

Question 1: current question
Question 2: not answered
Question 3: not answered

Generate terms from a term-to-term rule

A sequence starts at 2 with term-to-term rule “add 3.5”. The third term is

Practise

Topical questions

18 exam-style questions with model answers and mark schemes.

A set of structured and calculation questions matched to this chapter, in three tiers, each with a model answer, mark scheme and examiner tip. Higher-tier questions are tagged.

Practise topical questions

Exam practice

Practice paper

A 30-mark paper in the style of a GCSE paper, with a separate mark scheme. Give yourself 40 min.

Question paper

Chapter 10 Practice Paper

30 marks · 40 min · 5 pages

10 multiple-choice questions and structured questions with answer lines. Chapter 10: term-to-term and position-to-term rules, arithmetic nth terms and membership, triangular, square and cube numbers, Fibonacci-type and geometric sequences (Higher), and quadratic nth terms (Higher).

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Mark scheme

PRO

Chapter 10 Practice Paper — mark scheme

Answers · Marking points · 3 pages

Model answers with a mark for each creditworthy point — for marking at home or in class.

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How is this chapter examined?

Paper 1

non-calculator · 1 h 30

Exact fractions, surds and multiples of π expected

Papers 2 and 3

calculator · 1 h 30 each

Non-exact answers to 3 s.f., angles to 1 d.p.

Tiers

Foundation (grades 1–5) · Higher (grades 4–9)

Higher assumes all Foundation content

Want a full paper? Try the Paper 1-style (Foundation), Papers 2 & 3 style (Foundation), Paper 1-style (Higher), Papers 2 & 3 style (Higher) sets.

Questions

Frequently asked questions

Which parts of this chapter are Higher only?

Generating terms, linear nth terms and the special sequences (triangular, square, cube) appear on both tiers. Fibonacci-type sequences, geometric progressions, quadratic nth terms and surd ratios are Higher-tier.

How do I test whether a number is in a sequence?

Set the nth term equal to the number and solve. If the position n comes out as a positive whole number, the value is a term; if it is fractional (or negative), it never appears. The solving line is what earns the mark.

What is the trick for quadratic sequences?

Check the second differences are constant, halve that constant for the n² coefficient, then subtract your n² term from every term in the sequence. What remains is a linear sequence whose nth term finishes the job.

Why are positions numbered from 1?

GCSE sequences start at the first term, n = 1 — not n = 0. Always check your nth term against the first and second given terms, because a rule that only works from n = 0 will fail every real term.

GCSE Maths Sequences & nth term revision