Sequences follow patterns that can be described using rules such as the nth term. Recognising and forming these rules is an important algebra skill.
A sequence is a list of numbers that follow a rule or pattern.
Each number in the list is called a term.
In GCSE Maths, you need to continue sequences, spot the pattern, and sometimes find a formula for the nth term.
A list of numbers that follow a rule or pattern.
Each individual number in a sequence.
The number that appears first in the sequence.
The amount added or subtracted each time in an arithmetic sequence.
A sequence where the difference between consecutive terms is always the same.
A formula that gives any term in the sequence.
A sequence with a constant first difference.
A sequence where the difference does not stay the same.
Subtract consecutive terms to spot the pattern.
If the difference stays the same, the sequence is arithmetic.
For linear sequences, nth term is usually in the form \( an + b \).
Test your nth term with the first few term numbers.
\( 4,\ 7,\ 10,\ 13,\dots \)
\( 20,\ 16,\ 12,\ 8,\dots \)
\( 1,\ 4,\ 9,\ 16,\dots \)
\( 1,\ 3,\ 6,\ 10,\dots \)
A sequence is an ordered list of numbers. Each term follows a rule or pattern.
Check how much each term changes by.
Use the pattern to find the next terms.
The nth term gives a formula for any position in the sequence.
If the difference is not constant, the sequence is not linear.
Exam-style questions inspired by Edexcel GCSE Mathematics.
Write down the next two terms in the sequence: 3, 7, 11, 15, ...
Find the next term in the sequence: 2, 6, 18, 54, ...
Find the nth term of the sequence: 4, 7, 10, 13, ...
Find the 20^{\text{th}} term of the sequence with nth term \( 3n + 2 \).
Is 95 a term in the sequence defined by \( 5n - 5 \)? Show your working.
Exam-style questions based on the AQA GCSE Mathematics specification, focusing on algebraic reasoning and forming rules.
Find the nth term of the sequence: 5, 9, 13, 17, ...
Find the nth term of the sequence: 2, 5, 10, 17, 26, ...
The nth term of a sequence is \( 4n - 1 \). Write down the first four terms.
The nth term of a sequence is \( n^2 + 2 \). Find the 10th term.
Explain why the sequence defined by \( 2n + 3 \) will never contain the number 50.
Exam-style questions aligned with OCR GCSE Mathematics, emphasising reasoning, patterns, and real-life applications.
Here is a sequence: 1, 4, 9, 16, 25, ... Write down the next term and state the rule.
Find the nth term of the sequence: 3, 8, 15, 24, 35, ...
A pattern is made using matchsticks. The number of matchsticks in each stage forms the sequence 4, 7, 10, 13, ... Find a formula for the nth term.
Given that the nth term of a sequence is \( 2n^2 - n \), find the 5^{\text{th}} term.
Determine whether 132 is a term in the sequence defined by \( n^2 + n \). Show your reasoning.
Look carefully at how the sequence changes from one term to the next.
Find the common difference if the sequence is linear.
Use the difference to build the nth term.
Check your nth term with term 1, term 2 and term 3.
A small subtraction error can make the whole nth term wrong.
Students often choose the right multiple of \(n\) but add or subtract the wrong number.
Always test the nth term on the first few terms.
Some sequences are non-linear, so a linear rule will not work.
These are common mistakes students make when working with sequences in GCSE Maths.
A student calculates the step between terms incorrectly.
Find the difference between consecutive terms carefully. Check more than one pair to confirm the pattern.
A student substitutes \(n = 0\) for the first term.
In GCSE sequences, the first term usually corresponds to \(n = 1\), not 0.
A student finds the correct multiple of \(n\) but adds or subtracts the wrong number.
After finding the multiple (from the common difference), compare with the first term to determine the correct adjustment.
A student applies a linear formula to a non-linear sequence.
Check whether the differences are constant. If not, the sequence may be quadratic or follow a different pattern.
A student writes an nth term without verifying it.
Substitute values of \(n\) into your formula to check it produces the given terms.
Practise identifying and extending number sequences.
Find patterns and continue sequences.
What is the next number in the sequence: 3, 6, 9, 12, ... ?
Find the next term: 5, 10, 15, 20, ...
What is the next number: 20, 18, 16, 14, ... ?
Find the next term: 7, 14, 21, ...
Which sequence is increasing by 4 each time?
Find the next term: 50, 45, 40, ...
A student says the sequence 3, 6, 12, 24 increases by 3. What is the mistake?
Find the next term: 1, 4, 7, 10, ...
Which sequence is decreasing?
Find the next term: 100, 90, 80, ...
Find rules and nth terms of sequences.
Find the nth term of: 2, 5, 8, 11, ...
Find the nth term of: 4, 7, 10, 13, ...
Find the nth term: 5, 9, 13, 17, ...
Find the nth term: 3, 6, 9, 12, ...
Find the 10th term of: 2, 5, 8, 11, ...
Find the 6th term of: 4, 7, 10, 13, ...
A student says the nth term of 2, 5, 8, 11 is \(3n + 2\). What is wrong?
Find the nth term: 7, 12, 17, 22, ...
Which nth term matches: 10, 8, 6, 4, ... ?
Find the nth term: 1, 3, 5, 7, ...
Practise this topic with interactive games.
A list of numbers that follow a pattern.
A formula that gives any term in the sequence.
Look at how the sequence changes and relate it to position.