AQA GCSE Mathematics Paper 3 (Foundation), June 2025: Question 20

3 marks · Medium difficulty · Multi-step Problem

Find the nth term of a linear sequence given that the second term is 6 and the fifth term is 18.

Practise this question

Question

Question 20 states: A linear sequence has 2nd term = 6, 5th term = 18. Work out the nth term of the sequence. There are blank working lines and an answer line at the bottom, worth 3 marks.

Mark scheme

Show the mark scheme Mark scheme table for question 20 shows: M1 for (18 - 6) ÷ 3 or 4, or implied by sequence values 10 and 14; A2 for 4n - 2 (with A1 for 4n + c oe). Additional guidance specifies condoning alternative variable letters such as N, condoning n = 4n - 2, and scoring rules for different representations of the algebraic expression.

How to answer it

Linear Sequences: Finding the nth Term

📌 What this question tests

This question assesses your ability to find the general rule (nth term) of an arithmetic (linear) sequence when given two non-consecutive terms:

  • Calculating the common difference across non-consecutive positions.
  • Determining the zeroth term (constant offset, c).
  • Expressing a sequence rule using standard algebraic notation ( dn + c ).

Question 20 Walkthrough

A linear sequence has 2nd term = 6 and 5th term = 18. Work out the nth term. [3 marks]

📐 Step-by-Step Calculation

  1. Find the common difference (d):
    Difference in values: 18 - 6 = 12
    Difference in term positions: 5 - 2 = 3 steps
    Common difference = 12 ÷ 3 = 4
  2. Find the first term (n = 1):
    1st term = 2nd term - 4 = 6 - 4 = 2
    (The sequence is: 2, 6, 10, 14, 18...)
  3. Find the zeroth term (c):
    Zeroth term = 1st term - 4 = 2 - 4 = -2
  4. Write down the rule:
    Rule = dn + c = 4n - 2

✅ Correct Answer & Marks

4n - 2

Mark Breakdown:
  • M1: For (18 - 6) ÷ 3 or finding the difference 4 (also implied by writing the missing terms 10 and 14 ).
  • A1: For an expression of the form 4n + c (correct coefficient of n).
  • A2: For the fully correct simplified expression 4n - 2 .

💡 Key Knowledge

  • Linear Sequence Form: The nth term of any arithmetic sequence is written as an + b , where a is the term-to-term rule (common difference) and b is the "term 0" value.
  • Non-consecutive Jump: From the 2nd term to the 5th term is 5 - 2 = 3 additions of the common difference, not 2 or 5.

🧠 Exam Technique & Checking

  • Always verify with a given term: Substitute n = 5 into your final answer:
    4(5) - 2 = 20 - 2 = 18 . It matches the question!
  • Fill in the blanks: Sketch out the positions:
    __ , 6 , __ , __ , 18
    Writing down 10 and 14 directly earns the M1 mark even if you get stuck later.

❌ Common Errors & Traps

  • Dividing by the wrong number: Calculating (18 - 6) ÷ 2 = 6 by mistakenly assuming there are only 2 gaps between the 2nd and 5th term.
  • Writing n + 4: Confusing the common difference with the term formula. The mark scheme specifically awards M1 A0 for n + 4 .
  • Non-standard algebra: Expressions like 4 × n - 2 or n4 - 2 lose the final accuracy mark (A2).
  • Unsimplified signs: Writing 4n + -2 only earns A1; it must be simplified to 4n - 2 .

Topics

Algebra · 3.2.4 Sequences

Question and mark scheme from the AQA GCSE Mathematics examination, Paper 3 (Foundation), June 2025. QuestionVault is an independent revision resource; questions remain the copyright of the awarding body.