AQA GCSE Mathematics Paper 3 (Higher), June 2025: Question 8

3 marks · Medium difficulty · Multi-step Problem

Work out the nth term of a linear sequence given that the 2nd term is 6 and the 5th term is 18.

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Question

Question 8 asks: 'A linear sequence has 2nd term = 6 and 5th term = 18. Work out the nth term of the sequence.' Lines are provided for working and an answer line at the bottom. It is worth 3 marks.

Mark scheme

Show the mark scheme Mark scheme for Question 8 shows: M1 for (18 - 6) / 3 or 4 (or implied by values 10 and 14); A2 for 4n - 2 (with A1 for 4n + c oe). Additional guidance clarifies that using N is condoned, n = 4n - 2 is awarded M1A2, 4n + -2 is M1A1, and 4n written as 4 x n or n4 gets M1A1 but not M1A2.

How to answer it

Linear Sequences: Finding the nth Term

📋 What This Question Tests
  • Finding the common difference (term-to-term rule) between non-consecutive terms of an arithmetic sequence.
  • Writing an algebraic expression for the nth term rule in the general form an + b .
  • Working backwards to calculate the "zeroth term" ( n = 0 ) or solving algebraically.
  • Correct algebraic notation and simplification (avoiding expressions like n4 or 4n + -2 ).

Question 8 (3 Marks)

A linear sequence has 2nd term = 6 and 5th term = 18. Work out the nth term of the sequence.

📐 Step-by-Step Solution

1 Find the number of step intervals:
Between the 2nd term and the 5th term, there are 5 - 2 = 3 equal common differences ( d ).

2 Calculate the common difference ( d ):
The difference in value is 18 - 6 = 12 .
Therefore: 3d = 12 → d = 12 ÷ 3 = 4 .
The sequence increases by +4 each term.

3 List the sequence to verify:
• 1st term = 6 - 4 = 2
• 2nd term = 6
• 3rd term = 10
• 4th term = 14
• 5th term = 18

4 Find the constant term (zeroth term / adjustment):
Since the common difference is 4, the expression starts with 4n .
When n = 1 : 4(1) = 4 . To get the first term ( 2 ), subtract 2 : 4 - 2 = 2 .
Alternatively, the "zeroth term" is 1st term - 4 = 2 - 4 = -2 .

5 Form the final nth term:
4n - 2

✅ Correct Answer & Mark Breakdown

Answer: 4n - 2

M1: For (18 - 6) ÷ 3 or finding the difference 4 (also implied by writing terms 10 and 14 ).
A1: For finding any expression of the form 4n + c (where c is an integer or constant).
A2: For fully correct expression 4n - 2 .

💡 Key Knowledge

  • Linear sequence: A sequence with a constant first difference between consecutive terms.
  • General nth term formula: dn + c , where d is the common difference, and c is the term before the first term (the "zeroth" term, n = 0 ).
  • Gap between term p and term q is (q - p) × d .

🧠 Exam Technique & Examiner Advice

  • Always check by substituting back: Test your formula with n = 5 :
    4(5) - 2 = 20 - 2 = 18 . This matches the question perfectly!
  • Write out missing terms: Filling in the spaces between term 2 ( 6 ) and term 5 ( 18 ) as 6, 10, 14, 18 guarantees you the method mark ( M1 ) even if your final algebra goes wrong.
  • Algebraic formatting: Write 4n , not n4 or 4 × n . While examiners condone writing n = 4n - 2 , always write the expression cleanly as 4n - 2 .

❌ Common Errors to Avoid

  • Dividing by the wrong gap: Dividing 18 - 6 by 2 or 5 instead of 3 (the difference between term positions 5 - 2 = 3 ).
  • Writing the term-to-term rule: Giving n + 4 or "+4 each time" instead of the position-to-term rule ( 4n - 2 ). Note: n + 4 only gets M1A0 .
  • Unsimplified signs: Writing 4n + -2 loses the final accuracy mark ( M1A1 only ). Always simplify to 4n - 2 .
  • Using the wrong starting term: Forgetting that 6 is the 2nd term, not the 1st term, leading to an incorrect constant like 4n + 2 .

Topics

Algebra · 3.2.4 Sequences

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