GCSE Maths · Algebra

Quadratic Sequences GCSE Questions and Worked Answers

A quadratic sequence has a constant second difference and an nth term containing n². Half the second difference gives the coefficient of n²; subtract that square-number pattern to reveal the remaining linear rule.

Edexcel · AQA · OCRFoundation recognition · Higher nth term15 original questions

Ask Ari about quadratic sequences Start the questions

Free AI tutor · GCSE quadratic sequences

Practise GCSE quadratic sequences for free with an AI tutor

Ask Ari for an explanation or work through an original exam-style question together.

Hi, I’m Ari. I can explain second differences, help you build an nth term or check a sequence calculation line by line.

Ari is an AI tutor and can make mistakes. Use the worked answers below to check important results.

Look at how the changes change

What you need to know about quadratic sequences

A sequence is an ordered list of numbers. The difference tells you how much one term changes to the next. In a linear sequence that difference stays fixed. In a quadratic sequence the first differences change, but the differences between those differences — the second differences — stay fixed.

Terms → differences → nth term

See the second difference appear

Start with 3, 8, 15, 24, 35. Subtract neighbouring terms once, then subtract neighbouring first differences.

Terms3, 8, 15, 24, 35an ordered number pattern

First differences+5, +7, +9, +11these are not constant

Second differences+2, +2, +2constant, so the pattern is quadratic

The first differences change, but the second differences stay at +2. That constant second difference is the quadratic pattern.

From problem to method

Typical problems you need to be able to solve

These are the main kinds of question in this topic. Each card first states the problem, then explains how to solve it.

Continue a quadratic sequence

What the problem asks: Find later terms when a sequence has changing first differences and a constant second difference.

How to solve it: Continue the constant second-difference row, use it to extend the first differences, then add those differences to the terms.

$\text{terms}\to\Delta_1\to\Delta_2\text{ (constant)}$

Confirm a sequence is quadratic

What the problem asks: Decide whether a list follows a quadratic rather than linear pattern.

How to solve it: Calculate first and second differences. A non-zero constant second difference is the evidence of a quadratic sequence.

$\Delta_2=\text{constant}\ne0$

Find the nth term

What the problem asks: Write a rule in n for a quadratic sequence.

How to solve it: Use half the second difference for the n² coefficient, subtract that square-number sequence and find the linear nth term left over.

$u_n=an^2+bn+c,\qquad a=\frac{\Delta_2}{2}$

Generate terms from a rule

What the problem asks: Find terms when an nth-term formula such as 2n² − 3n + 4 is given.

How to solve it: Substitute n = 1, 2, 3 and so on, square before multiplying, and keep negative signs attached to their terms.

$u_3=2(3)^2-3(3)+4$

Test whether a number is a term

What the problem asks: Decide whether a stated value appears in a sequence with a known nth term.

How to solve it: Set the nth-term expression equal to the value, solve the quadratic and accept only positive whole-number positions.

$an^2+bn+c=\text{stated value}$

A reliable routine

Method for finding an² + bn + c

Use this routine when a sequence has a constant second difference and the question asks for its nth term. The values an² are a, 4a, 9a, 16a; their first differences are 3a, 5a, 7a, so every second difference is 2a.

Calculate the first differences, then the second differences.
Divide the constant second difference by 2 to find a.
Write the values of an² for n = 1, 2, 3, … and subtract them from the original terms.
Find the linear nth term bn + c of the remaining sequence.
Combine the parts and substitute n = 1 and n = 2 to check the original terms.

Check: You can check the coefficients directly: first difference = 3a + b, and first term = a + b + c. If the remainder is not linear, recheck a and your subtraction.

Fully worked

Quadratic sequences GCSE worked examples

Each solution explains the clue, why the method fits and how to check the result.

Example 1

Continue from the differences

2 marks

Question

Find the next two terms of

$4,\ 9,\ 16,\ 25,\ldots$

The first differences are

$+5,\ +7,\ +9$

They increase by $2$ , so the next differences are $+11$ and $+13$ .

$25+11=36$

$36+13=49$

$\boxed{36,\ 49}$

Example 2

Identify the quadratic pattern

2 marks

Question

Show that $6,11,18,27,38$ is a quadratic sequence.

First differences:

$5,\ 7,\ 9,\ 11$

Second differences:

$2,\ 2,\ 2$

The non-zero second difference is constant, so the sequence is $\boxed{\text{quadratic}}$ .

Example 3

Find an nth term with a = 1

Higher only4 marks

Question

Find the nth term of

$4,\ 11,\ 20,\ 31,\ 44,\ldots$

The second difference is $2$ , so

$a=2\div2=1$

Subtract $n^2$ and keep each remainder under its position.

$\begin{array}{c|ccccc}n&1&2&3&4&5\\\hline u_n&4&11&20&31&44\\n^2&1&4&9&16&25\\u_n-n^2&3&7&11&15&19\end{array}$

The remainder increases by $4$ , so its nth term is $4n-1$ .

$\boxed{n^2+4n-1}$

Example 4

Find an nth term with a = 2

Higher only4 marks

Question

Find the nth term of

$5,\ 12,\ 23,\ 38,\ 57,\ldots$

The first differences are $7,11,15,19$ , so the second difference is $4$ .

$a=4\div2=2$

Subtract $2n^2$ and organise the values by position.

$\begin{array}{c|ccccc}n&1&2&3&4&5\\\hline u_n&5&12&23&38&57\\2n^2&2&8&18&32&50\\u_n-2n^2&3&4&5&6&7\end{array}$

This remainder has nth term $n+2$ .

$\boxed{2n^2+n+2}$

Example 5

Handle a negative second difference

Higher only4 marks

Question

Find the nth term of

$10,\ 15,\ 18,\ 19,\ 18,\ldots$

The first differences are $5,3,1,-1$ , so the second difference is $-2$ .

$a=-2\div2=-1$

Subtract $-n^2$ by adding $n^2$ to each original term.

$11,\ 19,\ 27,\ 35,\ 43$

The remainder has nth term $8n+3$ .

$\boxed{-n^2+8n+3}$

Example 6

Generate terms from a formula

Higher only3 marks

Question

Find the first four terms of $2n^2-3n+4$ .

Substitute $n=1,2,3,4$ .

$n=1:\quad2(1)^2-3(1)+4=3$

$n=2:\quad2(2)^2-3(2)+4=6$

$n=3:\quad2(3)^2-3(3)+4=13$

$n=4:\quad2(4)^2-3(4)+4=24$

$\boxed{3,\ 6,\ 13,\ 24}$

Example 7

Find a distant term

Higher only2 marks

Question

The nth term is $n^2+3n-2$ . Find the 20th term.

$20^2+3(20)-2$

$=400+60-2$

$\boxed{458}$

Example 8

Test whether a value is a term

Higher only5 marks

Question

The nth term of a sequence is $n^2+2n$ . Is $168$ a term of the sequence?

Set the nth term equal to $168$ .

$n^2+2n=168$

$n^2+2n-168=0$

$(n+14)(n-12)=0$

$n=-14\quad\text{or}\quad n=12$

A position must be a positive whole number, so $n=12$ is valid.

$\boxed{168\text{ is the 12th term}.}$

15 original questions · total 52 marks

Quadratic sequences GCSE exam-style questions

Try each question before opening its fully worked answer. The difficulty rises through the set.

Before you startAllow about 60 minutes · show both difference rows and check your nth term against at least two original terms · answers start collapsed

Continue a sequence

2 marks

Find the next two terms: $2,6,12,20,\ldots$

Show worked answer

First differences are $4,6,8$ , increasing by $2$ . The next differences are $10$ and $12$ .

$20+10=30$

$30+12=42$

$\boxed{30,\ 42}$

Calculate second differences

2 marks

Find the constant second difference of $7,16,29,46,67$ .

Show worked answer

First differences are $9,13,17,21$ .

Second differences are $4,4,4$ .

$\boxed{4}$

Find the n² coefficient

Higher only2 marks

A quadratic sequence has constant second difference $10$ . Find the coefficient of $n^2$ .

Show worked answer

The second difference is $2a$ .

$a=10\div2=\boxed{5}$

Find a simple nth term

Higher only4 marks

Find the nth term of $2,8,18,32,50,\ldots$

Show worked answer

The second difference is $4$ , so $a=2$ .

The terms are exactly $2n^2$ :

$2(1)^2,\ 2(2)^2,\ 2(3)^2,\ldots$

$\boxed{2n^2}$

Find n² plus a constant

Higher only4 marks

Find the nth term of $6,9,14,21,30,\ldots$

Show worked answer

The second difference is $2$ , so start with $n^2$ .

Subtract $1,4,9,16,25$ from the terms.

$5,5,5,5,5$

$\boxed{n^2+5}$

Find n² plus a linear term

Higher only4 marks

Find the nth term of $5,14,25,38,53,\ldots$

Show worked answer

The second difference is $2$ , so start with $n^2$ .

Subtracting $n^2$ leaves

$4,10,16,22,28$

This has nth term $6n-2$ .

$\boxed{n^2+6n-2}$

Use a larger second difference

Higher only4 marks

Find the nth term of $8,19,36,59,88,\ldots$

Show worked answer

First differences are $11,17,23,29$ , so the second difference is $6$ .

$a=6\div2=3$

Subtracting $3n^2$ leaves $5,7,9,11,13$ , whose nth term is $2n+3$ .

$\boxed{3n^2+2n+3}$

Use a negative quadratic coefficient

Higher only4 marks

Find the nth term of $12,19,22,21,16,\ldots$

Show worked answer

First differences are $7,3,-1,-5$ , so the second difference is $-4$ .

$a=-4\div2=-2$

Subtracting $-2n^2$ leaves $14,27,40,53,66$ , whose nth term is $13n+1$ .

$\boxed{-2n^2+13n+1}$

Generate five terms

3 marks

Find the first five terms of $n^2-n+3$ .

Show worked answer

Substitute $n=1,2,3,4,5$ .

$\boxed{3,\ 5,\ 9,\ 15,\ 23}$

Find the 15th term

2 marks

Find the 15th term of $2n^2+n-4$ .

Show worked answer

$2(15)^2+15-4$

$=450+11$

$\boxed{461}$

Write an nth term from a tile pattern

Higher only3 marks

Stage $n$ contains an $n$ by $n$ square and one extra row of $n$ tiles. Write an expression for the number of tiles in stage $n$ .

Each stage contains an n × n square plus one extra row of n tiles, so the structure itself gives n² + n.

Show worked answer

The square contains $n\times n=n^2$ tiles.

The extra row contains $n$ tiles.

$\boxed{n^2+n}$

Test a value

Higher only4 marks

Is $80$ a term of the sequence with nth term $n^2+n$ ?

Show worked answer

Check the whole-number positions around $80$ .

$u_8=8^2+8=72$

$u_9=9^2+9=90$

The sequence is increasing for positive $n$ , and $80$ lies strictly between consecutive terms $72$ and $90$ .

$\boxed{80\text{ is not a term}.}$

Find the position of a term

Higher only4 marks

Which term of $n^2+5n$ is $66$ ?

Show worked answer

$n^2+5n=66$

$n^2+5n-66=0$

$(n+11)(n-6)=0$

The positive whole-number solution is $n=6$ .

$\boxed{66\text{ is the 6th term}.}$

Find an unknown first term

4 marks

The sequence $k,19,34,55,82$ has constant second difference $6$ . Find $k$ .

Show worked answer

Known first differences are $15,21,27$ , so the previous first difference must be $9$ because these rise by $6$ .

$19-k=9$

$\boxed{k=10}$

Derive and use an nth term

Higher only6 marks

The sequence is $1,10,27,52,85,\ldots$

(a) Find its nth term.

(b) Find the 12th term.

Show worked answer

First differences are $9,17,25,33$ , so the second difference is $8$ .

$a=8\div2=4$

Subtracting $4n^2$ leaves

$-3,-6,-9,-12,-15$

The remainder is $-3n$ .

$\text{nth term}=4n^2-3n$

For $n=12$ ,

$4(12)^2-3(12)=576-36$

$\boxed{540}$

Examiner-style feedback

Common quadratic sequences mistakes

Stopping at first differences

Changing first differences do not prove there is no pattern. Calculate the differences between them.

Using the second difference as a

For an², the second difference is 2a. Divide it by 2.

Subtracting n² when a is not 1

Subtract the complete an² sequence: if a = 3, use 3, 12, 27, 48, …

Using a first difference as the coefficient of n

The first differences change. For an², use half the constant second difference to find a.

Accepting a non-integer position

A term position n must be a positive whole number.

30-second recap

Choose, carry out, check

Translate the wording before you reach for a rule.

First differences measure each change.
A constant non-zero second difference signals a quadratic sequence.
Half the second difference gives a.
Subtract an², find the linear remainder and check.

Quick answers

Quadratic sequences FAQ

What is a quadratic sequence?

It is a sequence whose nth term contains n² and whose non-zero second differences are constant.

Why do I halve the second difference?

The second difference of an² is 2a, so dividing by 2 recovers the coefficient a.

Can a quadratic sequence go down?

Yes. A negative n² coefficient gives a negative constant second difference and the sequence may eventually decrease.

How do I check an nth term?

Substitute n = 1, n = 2 and another position. The outputs should match the original terms.

Build connected skills

What to revise next

Power rules

Indices

Strengthen calculations involving n² and other powers.

Revise indices

Quadratic algebra

Completing the square

Rewrite quadratic expressions and solve equations.

Revise completing the square

Substitution

Functions

Evaluate algebraic rules for chosen inputs.

Revise functions

Content standards

Curriculum and rights review

Reviewed 5 September 2026 against DfE content A23–A25, Pearson Edexcel A23–A25, AQA A23–A25 and OCR J560 sections 6.06a–b. All questions, patterns and tables are original.

Official specification references