AQA GCSE Mathematics Paper 3 (Higher), June 2025: Question 18

Finding the nth Term of a Quadratic Sequence

📋 What this question tests

This question assesses your ability to find an algebraic expression for the nth term of a quadratic sequence in the form an² + bn + c . You need to calculate first and second differences, determine the coefficient of n² by halving the second difference, and find the remaining linear sequence to deduce b and c .

Question 18 (3 Marks)

AQA GCSE Mathematics (Higher Tier)

📐 Step-by-Step Method

Given sequence: 6, 15, 28, 45

Step 1: Find the 1st and 2nd differences

Terms	6	15	28	45
1st Difference	+9 +13 +17
2nd Difference	+4 +4

Step 2: Find the coefficient of n² ( a )

Coefficient a = (2nd difference) ÷ 2
a = 4 ÷ 2 = 2 , so the leading term is 2n².

Step 3: Subtract 2n² from the original terms

n	1	2	3	4
Original Sequence	6	15	28	45
2n² (2, 8, 18, 32)	2	8	18	32
Difference (Original − 2n²)	4	7	10	13

Step 4: Find the nth term of the remaining linear sequence

Linear sequence: 4, 7, 10, 13...
Common difference = +3 → 3n
Zero term (or 4 − 3) = +1
Linear part = 3n + 1

Step 5: Combine into final expression

nth term = 2n² + 3n + 1

✅ Correct Answer & Mark Scheme

2n² + 3n + 1

Or explicitly stating: a = 2, b = 3, c = 1

Mark Breakdown:

B3: Correct fully simplified expression: 2n² + 3n + 1 .
B2: Two correct components found:
- 2n² + 3n (+ c) or a = 2 and b = 3
- OR 2n² (+ bn) + 1 or a = 2 and c = 1
B1: One correct component found:
- 2n² (+ bn + c) or a = 2
- OR an² (+ bn) + 1 (where a ≠ 0) or c = 1
- OR states/shows that second difference = 4

💡 Key Knowledge

Quadratic form: an² + bn + c .
Constant 2nd difference: In any quadratic sequence, the second difference between consecutive terms is always constant and equal to 2a .
Alternative Formula Method:
- 2a = 2nd difference
- 3a + b = 1st difference (between term 1 and term 2)
- a + b + c = 1st term
Using this question:
• 2a = 4 → a = 2
• 3(2) + b = 9 → 6 + b = 9 → b = 3
• 2 + 3 + c = 6 → 5 + c = 6 → c = 1

🧠 Exam Technique & Examiner Tips

Always check by substituting: Substitute n = 1 and n = 2 back into your final expression.
• For n = 1: 2(1)² + 3(1) + 1 = 2 + 3 + 1 = 6 ✓
• For n = 2: 2(2)² + 3(2) + 1 = 8 + 6 + 1 = 15 ✓
Write down differences clearly: Even if your final algebra goes wrong, simply identifying and writing that the second difference is 4 guarantees you B1 !
Order of terms doesn't matter: Writing 1 + 3n + 2n² receives full marks, but standard form an² + bn + c avoids silly sign mistakes.

❌ Common Errors & Traps

Forgetting to halve the 2nd difference: A very frequent error is using 4 as the coefficient to write 4n² instead of 2n² .
Writing an equation instead of an expression: Writing 2n² + 3n + 1 = 0 drops a mark (penalised to B2). Do not include = 0 .
Using the wrong variable: Using x instead of n (e.g. 2x² + 3x + 1 ) can lose precision marks. Always use n as requested.
No working shown with incorrect values: Simply writing an answer like 5n + 4 without showing that 4 came from the second difference scores 0 marks .

AQA GCSE Mathematics Paper 3 (Higher), June 2025: Question 18

Question

Mark scheme

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