Find the nth term an² + bn + c of a sequence whose second differences are constant, and use it to test or find terms. It is Higher only, aimed at grade 8. Free questions with answers are on this page.
A quadratic sequence has a constant second difference. That second difference is 2a in an2+bn+c. So a=2second difference. Subtract an2 from each term to leave a linear sequence, then find b and c as usual. Check the first three terms.
An alternative is to plug n=1,2,3 into an2+bn+c and solve the three simultaneous equations. The nth term of 2,6,12,20,… is n2+n (first differences 4,6,8, second difference 2, so a=1). Finding a term number means solving a quadratic inequality or equation in n and keeping the positive integer root.
Objectives
Halve the second difference to get the n^2 coefficient.
Find the rest of the term by comparing.
Use the term to find a position.
Key points
Recognise a quadratic sequence: the first differences change but the second differences are constant.
Find a by halving the constant second difference.
Subtract an2 from each term and find the linear nth term of what is left.
Check the final formula against at least two terms of the sequence.
Worked examples
Example 1
Find the nth term of 2, 6, 12, 20.
2 marks
First differences 4, 6, 8. Second difference 2
Half of 2 is 1, so start with n^2
Leftover 1, 2, 3, 4 so the term is n^2+n
Answer: n^2+n
Common slips
Slip: First differences are the terms themselves Subtract consecutive terms.
Slip: Use 2n^2 Halve the second difference.
Slip: The term is n^2 The leftovers are the extra +n.
Example 2
Find the nth term of 1, 6, 15, 28.
2 marks
Second difference 4
Coefficient of n^2 is 2
2n^2-n
Answer: 2n^2-n
Common slips
Slip: Second difference 1 9-5=4 and 13-9=4.
Slip: Coefficient is 4 Half of 4 is 2.
Slip: 2n^2 The linear part is -n. Term 1 is 2-1=1.
Example 3
The nth term is n^2+n. Which term equals 30?
3 marks
n^2+n-30=0
(n+6)(n-5)=0
n=5
Answer: 5
Common slips
Slip: n=30 Set the expression equal to 30.
Slip: (n-6)(n-5)=0 6 and -5 multiply to -30 and differ by 1.
Slip: n=-6 A position cannot be negative.
Questions and answers
GCSE Maths the nth term of a quadratic sequence questions and answers. Try each one, then open the answer. Easy questions come first.
Question 1 Easy2 marks
Give the nth term of the sequence 3, 8, 15, 24.
Show answer
Answer: n^2+2n
Second difference 2, so the square part is n^2.
Leftover is 2, 4, 6, 8.
M1 — Second difference 2, so the square part is n^2.
A1 — Answer n^2+2n
Question 2 Easy2 marks
Give the nth term of 4, 10, 18, 28.
Show answer
Answer: n^2+3n
Second difference 2.
Compare with n^2 and the leftover is 3n.
M1 — Second difference 2.
A1 — Answer n^2+3n
Question 3 Medium2 marks
Find the nth term of 2, 8, 18, 32.
Show answer
Answer: 2n^2
Second difference 4, so 2n^2.
The leftovers are zero.
M1 — Second difference 4, so 2n^2.
A1 — Answer 2n^2
Question 4 Medium2 marks
Which expression generates 3, 9, 19, 33?
Show answer
Answer: 2n^2+1
2n^2 matches 2, 8, 18, 32.
The original is 1 more each time.
M1 — 2n^2 matches 2, 8, 18, 32.
A1 — Answer 2n^2+1
Question 5 Hard2 marks
The nth term is n^2+2n. Which term number gives 24?
Show answer
Answer: 4
n^2+2n-24=0.
(n+6)(n-4)=0, and n is positive.
M1 — n^2+2n-24=0.
A1 — Answer 4
Question 6 Hard2 marks
The nth term is 2n^2. Which position has value 32?
Show answer
Answer: 4
2n^2=32 so n^2=16.
n=4, not -4.
M1 — 2n^2=32 so n^2=16.
A1 — Answer 4
Question 7 Exam-style5 marks
The nth term of a sequence is 2n^2-n. Find the 4th term, and the term number that equals 15.
Show answer
Answer: 28 and 3
Term 4 is 32-4=28.
2n^2-n=15.
2n^2-n-15=0.
(2n+5)(n-3)=0.
n=3.
M1 — Substitute 4
A1 — 28
M1 — Equation
M1 — Factor
A1 — n=3
Common mistakes
Most lost marks on this topic are one slip.
Using the second difference as the coefficient of n^2
The coefficient is half the second difference
Stopping after the n^2 part
Subtract it from the sequence and fit the linear part that is left
Keeping n = -6 when solving for a position
A term number must be a positive whole number
Spot the slip
The second difference is 2. A student uses n^2 as the whole nth term of 2, 6, 12, 20. What is missing?
The linear part n
The second difference should be squared
Sequences cannot be quadratic
Show the slip
The linear part n Half of 2 is 1, so the term starts n^2, but 1, 4, 9, 16 is not 2, 6, 12, 20. You still need +n.
Exam tips
Be ready to do this without a calculator. Paper 1 is the non-calculator paper.
If the question says “show that”, the result is given. The marks are for the steps that reach it.
The method is the same on AQA-style and Edexcel-style papers.
Lay out three rows: sequence, first differences, second differences. Clear layout earns method marks even if you slip later.
Check your formula against the 4th or 5th term, not just the 1st — the 1st term alone catches almost no errors.
Printable worksheet
The same questions are in a PDF you can print. Answers start on a later page.