Sequences GCSE Questions, Worked Examples and Answers

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Start with position and value

What you need to know about sequences

Imagine numbered places in a row. Position 1 contains the first term, position 2 the second term, and so on. A sequence rule connects each position to its value. For 4, 7, 10, 13, … the values rise by 3 each time, so its term-to-term rule is add 3.

Position → term → pattern

Connect the positions to the terms

The nth term 3n + 1 means: multiply the position n by 3, then add 1.

Position n1, 2, 3, 4where the term sits

Apply 3n + 14, 7, 10, 13multiply, then add

Difference+3 each timethe coefficient of n

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 sequence

What the problem asks: Write the next terms when the beginning of a sequence is shown.

How to solve it: Find the change from one term to the next and repeat it. If the changes themselves change, examine their pattern before continuing.

Generate terms from a rule

What the problem asks: Use a term-to-term instruction or an nth-term expression to list terms.

How to solve it: For a term-to-term rule, begin with the stated first term. For an nth term, substitute n = 1, 2, 3 and so on.

Find a linear nth term

What the problem asks: Write a formula for a sequence with a constant difference.

How to solve it: Use the common difference as the coefficient of n, then compare the generated first term with the actual first term to find the adjustment.

u_n=dn+c

Test whether a value is a term

What the problem asks: Decide whether a number occurs in a sequence with a known nth term.

How to solve it: Set the nth-term expression equal to the value. The value is a term only if the resulting position is a positive whole number.

Use a special sequence

What the problem asks: Continue or identify square, cube, triangular, Fibonacci-type, geometric or quadratic patterns.

How to solve it: Describe how the terms are built. Do not assume every sequence changes by adding the same amount.

A reliable routine

Method for a linear nth term

Use this method when the difference between neighbouring terms is constant. It works because increasing the position by 1 increases dn + c by d.

Subtract neighbouring terms to find the common difference d.
Start the rule with dn.
Substitute n = 1 into dn and compare it with the actual first term.
Add or subtract the difference needed to obtain the actual first term.
Check the rule at n = 2 and n = 3.

Check: A term-to-term rule is good for nearby terms. An nth-term rule is better for a distant position because it avoids listing every earlier term.

Fully worked

Sequences GCSE worked examples

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

Example 1

Continue an arithmetic sequence

2 marks

Question

Find the next two terms of $18,23,28,33,\ldots$

The change is $+5$ each time.

$33+5=38$

$38+5=43$

$\boxed{38,\ 43}$

Example 2

Generate terms from an nth term

3 marks

Question

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

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

$4(1)-3=1$

$4(2)-3=5$

$4(3)-3=9$

$4(4)-3=13$

$\boxed{1,\ 5,\ 9,\ 13}$

Example 3

Find a linear nth term

3 marks

Question

Find the nth term of $8,13,18,23,\ldots$

The common difference is $5$ , so begin with $5n$ .

At $n=1$ , $5n$ gives $5$ , but the sequence begins at $8$ . Add $3$ .

$\boxed{5n+3}$

Check: $5(2)+3=13$ .

Example 4

Test whether a value appears

3 marks

Question

The nth term is $6n-1$ . Is $83$ a term?

Set the rule equal to $83$ .

$6n-1=83$

$6n=84$

$n=14$

$14$ is a positive whole-number position, so $\boxed{83\text{ is the 14th term}}$ .

Example 5

Continue a Fibonacci-type sequence

2 marks

Question

Each term after the first two is the sum of the previous two. Continue $3,7,10,17,\ldots$ for two terms.

Add the latest two terms each time.

$10+17=27$

$17+27=44$

$\boxed{27,\ 44}$

10 original questions · total 26 marks

Sequences GCSE exam-style questions

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

Before you startAllow about 30 minutes · Show differences clearly and use a positive whole-number position when testing whether a value is a term. · answers start collapsed

1

Continue by adding

2 marks

Find the next two terms: $-4,2,8,14,\ldots$

Show worked answer

The common difference is $6$ .

$14+6=20,\qquad20+6=26$

$\boxed{20,\ 26}$

2

Continue by multiplying

2 marks

Find the next two terms: $5,15,45,135,\ldots$

Show worked answer

Each term is multiplied by $3$ .

$135\times3=405$

$405\times3=1215$

$\boxed{405,\ 1215}$

3

Use an nth term

3 marks

Find the first three terms of $7n+2$ .

Show worked answer

$7(1)+2=9$

$7(2)+2=16$

$7(3)+2=23$

$\boxed{9,\ 16,\ 23}$

4

Find an nth term

3 marks

Find the nth term of $3,9,15,21,\ldots$

Show worked answer

The difference is $6$ , so begin with $6n$ . This gives $6$ at $n=1$ , which is $3$ too large.

$\boxed{6n-3}$

5

Find an nth term with negatives

3 marks

Find the nth term of $12,8,4,0,\ldots$

Show worked answer

The difference is $-4$ , so begin with $-4n$ . At $n=1$ this gives $-4$ , so add $16$ .

$\boxed{-4n+16}$

6

Test a term

3 marks

Is $74$ a term of the sequence $4n+2$ ?

Show worked answer

$4n+2=74$

$4n=72$

$n=18$

Therefore $\boxed{74\text{ is the 18th term}}$ .

7

Reject a non-integer position

3 marks

Is $50$ a term of $6n+1$ ?

Show worked answer

$6n+1=50$

$6n=49$

$n=\frac{49}{6}$

This is not a whole-number position, so $\boxed{50\text{ is not a term}}$ .

8

Use square numbers

2 marks

Write the next two terms: $1,4,9,16,25,\ldots$

Show worked answer

These are $1^2,2^2,3^2,4^2,5^2$ .

The next terms are $6^2$ and $7^2$ .

$\boxed{36,\ 49}$

9

Use a Fibonacci-type rule

2 marks

Continue $2,5,7,12,19,\ldots$ for two terms.

Show worked answer

Each term is the sum of the previous two.

$12+19=31$

$19+31=50$

$\boxed{31,\ 50}$

10

Recognise a quadratic pattern

3 marks

Show that $2,6,12,20,30$ is quadratic.

Show worked answer

First differences are

$4,6,8,10$

Second differences are

$2,2,2$

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

Examiner-style feedback

Common sequences mistakes

Using the difference as the whole nth term

A difference of 5 means start with 5n; you still need to compare the first term and find the adjustment.

Beginning nth-term substitution at zero

For GCSE sequences, the first displayed term normally corresponds to n = 1 unless the question says otherwise.

Accepting a decimal position

A value occurs in the sequence only when solving for n gives a positive whole number.

Assuming every sequence is arithmetic

Check for multiplication, Fibonacci-type changes, familiar number patterns and changing differences.

30-second recap

Choose, carry out, check

Translate the wording before you reach for a rule.

A sequence is an ordered list.
Term-to-term rules move between neighbours.
Nth-term rules connect position directly to value.
For a linear sequence, the common difference is the coefficient of n.

Quick answers

Sequences FAQ

What is a term in a sequence?

A term is one value in the ordered list. Its position tells you whether it is the first, second, third or nth term.

What is the nth term?

It is a formula that gives the value at position n.

How do I know a sequence is quadratic?

Its first differences change but its second differences are constant and non-zero.

Can a sequence decrease?

Yes. Its common difference, multiplier or other rule can make later terms smaller.

Build connected skills

What to revise next

Changing differences

Quadratic sequences

Use second differences to derive a quadratic nth term.

Revise quadratic sequences

Linked unknowns

Simultaneous equations

Use elimination or substitution when two equations share two unknowns.

Revise simultaneous equations

Repeated multiplication

Compound interest

Connect geometric sequences to repeated percentage change.

Revise compound interest

Content standards

Curriculum and rights review

Reviewed 5 September 2026 against DfE content A23–A25, Pearson Edexcel, AQA and OCR specifications. The questions and explanations are original.

Official specification references