GCSE Maths • Foundation & Higher
GCSE Sequences Revision Guide
Recognise patterns, derive nth terms and calculate any required term.
Foundation
Exam practice
Worked examples
What You'll Learn
Continue a Sequence
Identify the intended rule and generate later terms accurately.
Find Linear Nth Terms
Use the common difference as the coefficient of n and calculate the adjustment.
Use Nth-Term Formulae
Substitute a position to find a term and solve an equation to test membership.
Recognise Geometric Sequences
Find a constant multiplier or divisor and write an exponential nth term.
Analyse Quadratic Sequences
Use first and second differences to identify and model a quadratic pattern.
Connect Sequences and Graphs
Plot (n, term) coordinates and relate linear or quadratic rules to graph shape.
Key Rules and Formulas
Scatter Graphs Lessons
Arithmetic Sequences
Use common differences to continue sequences and form linear nth-term rules.
Geometric Sequences
Use common ratios, powers and the formula ar^(n-1) for repeated multiplication.
Nth Term Sequences
Find, use and solve nth-term formulae for positions and membership.
Quadratic Sequences
Use constant second differences to derive quadratic nth-term rules.
CORE SUBTOPICS
Increasing and Decreasing Patterns
Handle positive, zero and negative common differences.
Difference Tables
Organise first and second differences accurately.
Growth and Decay
Use ratios greater than 1, between 0 and 1, or negative ratios where appropriate.
Special Sequences
Recognise square, cube, triangular and Fibonacci-style patterns when the rule is clear.
Position-to-Term Problems
Substitute the required positive integer position.
Term-to-Position Problems
Solve the rule and validate the resulting index.
Worked Example
Finding a Quadratic Nth Term
QUESTION
Find the nth term of 2, 7, 14, 23, 34, ...
- Given information: First differences are 5, 7, 9, 11; second differences are 2, 2, 2.
- Method: Use half the second difference for the n² coefficient, then compare and correct the residual sequence.
- Constant second difference 2 gives A = 2/2 = 1, so begin with n².
- The values of n² are 1, 4, 9, 16, 25.
- Subtract these from the original terms: 1, 3, 5, 7, 9.
- The residual sequence has nth term 2n - 1.
- Combine the parts: u_n = n² + 2n - 1.
- Check n=5: 25 + 10 - 1 = 34.
- Final answer: u_n = n² + 2n - 1.
- Independent answer check: Substituting n=1 to 5 reproduces 2, 7, 14, 23 and 34.
- One common mistake: Do not assume the rule is n²+n; that produces 2, 6, 12, 20, 30, not the given sequence.
- Common GCSE Mistakes
- Using Term Value as n
- Missing a Negative Difference
- Using n Instead of n-1 in Geometric Rules
- Stopping at the Second Difference
- Accepting a Non-Integer Position
- Joining Discrete Points Automatically
- Exam Tips
- Write term positions above the sequence before deriving a rule.
- Calculate differences or ratios systematically and keep signs exact.
- For a linear nth term, test n=1 and another position.
- For a quadratic sequence, show both difference rows.
- For membership, solve and then check that n is a positive integer.
- Use brackets carefully when substituting negative values or powers.
Ready to Practice?
Printable Worksheet
Download and print practice questions with answers.
Interactive Quiz
Test your knowledge with instant feedback and hints.
Exam-Style Questions
Timed questions to build confidence for the real exam.
Watch the Sequences Tutorial
Clear, step-by-step explanation of expanding brackets, simplifying complex expressions, and solving for x. Perfect for visual learners who need a breakdown of the rules in action.
- 12 minute comprehensive guide
- High-definition whiteboard walk-through
Continue Your GCSE Maths Revision
Continue Your GCSE Maths Revision
Frequently Asked Questions
A sequence is an ordered list of terms generated by a stated or inferred rule.
It is a formula for the term at position n, where n is normally a positive integer.
Use the common difference as the coefficient of n, then compare generated terms to find the constant adjustment.
Arithmetic sequences add a constant difference; geometric sequences multiply by a constant ratio.
Its first differences change but its second differences are constant and non-zero.
Set the nth-term rule equal to the number, solve, and check whether n is a positive integer.
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