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

Quadratic Sequences

In one sentence. Sequences whose differences themselves change — spotting the n² engine underneath and building the exact rule. Higher Tier.

Sequences — one of 2 modules in this strand, and one of 53 in Pearson Edexcel International GCSE Mathematics. Higher Tier only.

This module ends with Leonardo Fibonacci. Learn Quadratic Sequences, teach it, then defend it against Leonardo Fibonacci.

Where this sits in the specification

Specification 4MA1, 3.1 Sequences — quadratic sequences (Higher). 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 — quadratic sequences (Higher)

1 · Understand it

What it is

In a quadratic sequence the first differences change, but the SECOND differences are constant: 2, 6, 12, 20 has differences 4, 6, 8 and second difference 2. The rule contains n², and the n² coefficient is half the second difference.

Why it works

The sequence n² = 1, 4, 9, 16 has differences 3, 5, 7 — climbing by 2 every time. Any an² term produces a constant second difference of 2a, which is why halving the second difference recovers a. Subtracting the an² part from the sequence leaves a LINEAR leftover you already know how to solve.

When a question is asking for it

Whenever first differences aren't constant but second differences are. Growth patterns with area-like behaviour (tiles filling squares, handshake and diagonal counts) are quadratic by nature.

How to hold it in your head

Peel the onion: (1) second difference → the an² engine; (2) subtract the engine's values from the sequence; (3) the leftover is linear — solve it with the previous module's recipe; (4) glue: an² + leftover. Each layer is a technique you already own.

2 · The mistakes that cost marks

  • Giving up because differences 'aren't constant'. Take differences AGAIN — constancy one level down means quadratic.
  • Using the second difference itself as the n² coefficient. Halve it: second difference 2 → n², second difference 4 → 2n².
  • Forgetting to subtract the n² part before finding the linear bit — the leftover, not the original, gives the linear term.
  • Not testing the final rule on n = 1 and n = 2. Ten seconds, catches nearly everything.

3 · Worked examples

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

Find the nth term of 0, 5, 12, 21, 32, …

  1. First differences: 5, 7, 9, 11. Second difference: 2 → engine is n² (half of 2 is 1).
  2. Subtract n²: 0−1, 5−4, 12−9, 21−16, 32−25 → −1, 1, 3, 5, 7.
  3. Leftover is linear: difference 2, zeroth term −3, so 2n − 3.
  4. Rule: n² + 2n − 3. Test n = 3: 9 + 6 − 3 = 12 ✓

Answer: n² + 2n − 3

Find the nth term of 5, 12, 25, 44, …

  1. Differences: 7, 13, 19. Second difference: 6 → engine 3n².
  2. Subtract 3n²: 5−3, 12−12, 25−27, 44−48 → 2, 0, −2, −4.
  3. Leftover: difference −2, zeroth term 4, so 4 − 2n.
  4. Rule: 3n² − 2n + 4. Test n = 2: 12 − 4 + 4 = 12 ✓

Answer: 3n² − 2n + 4

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 — Work out the next term of the sequence 2, 6, 12, 20, …
  2. Grade 7 — Find the nth term of the sequence 4, 7, 12, 19, 28, … (Type it like n^2+3.)
  3. Grade 8 — Find the nth term of the sequence 3, 8, 15, 24, … (Type it like n^2+2n.)
  4. Grade 9 — Find the nth term of the sequence 2, 7, 16, 29, … (Type it like 2n^2-n+1.)
  5. 9+ — The nth term of the sequence 2, 6, 12, 20, … is n² + n. Which term of the sequence is equal to 210?

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 Quadratic Sequences, marked on the spot, that tell you what was missing rather than just what was wrong.
  2. Teach it — Explain how Quadratic Sequences 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 Quadratic Sequences against Leonardo Fibonacci — three rounds, marked against descriptors you can read before you begin.
  4. Then the exam — read all 10 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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Carry on through the course