Algebra Revision Notes

    Subject: Mathematics | Level: GCSE | Exam Board: Pearson

    Sequences are the heartbeat of algebraic patterns, testing your ability to spot relationships and express them mathematically. Mastering term-to-term and position-to-term rules is essential for unlocking high marks in your GCSE Mathematics exam.

    Revision Notes & Key Concepts

    ![GCSE Mathematics: Algebra - Sequences](https://xnnrgnazirrqvdgfhvou.supabase.co/storage/v1/object/public/study-guide-assets/guide_dbb772b2-585f-4e65-b9a3-d091bb57070f/header_image.png) ## Overview Sequences are a fundamental topic in GCSE Mathematics that bridge the gap between simple number patterns and complex algebra. A sequence is simply an ordered list of numbers, but the power lies in finding the rule that governs it. Examiners love sequences because they test multiple skills at once: pattern recognition, algebraic notation, and problem-solving. This topic connects heavily to linear graphs (where the common difference is the gradient) and functions. In your exam, you can expect a range of question styles. Foundation tier often focuses on generating terms from a rule or identifying special sequences. Higher tier demands finding the nth term of quadratic sequences or proving whether a specific number belongs to a sequence. Let's break it all down. ## Key Concepts ### Concept 1: Term-to-Term Rules A term-to-term rule tells you how to get from one number in the sequence to the very next number. You must know the previous term to find the next one. **Example**: The sequence 3, 7, 11, 15... has a term-to-term rule of "add 4". This is simple but limited. If an examiner asks for the 100th term, using a term-to-term rule would take forever! This is why we need position-to-term rules. ### Concept 2: Position-to-Term Rules (The nth Term) The "nth term" is an algebraic formula that links the position of a number (n) to its actual value. **Example**: If the nth term is $3n + 2$: - The 1st term ($n=1$) is $3(1) + 2 = 5$ - The 10th term ($n=10$) is $3(10) + 2 = 32$ This is incredibly powerful because it allows you to calculate any term instantly without knowing the previous ones. ![How to find the nth term of a linear sequence](https://xnnrgnazirrqvdgfhvou.supabase.co/storage/v1/object/public/study-guide-assets/guide_dbb772b2-585f-4e65-b9a3-d091bb57070f/nth_term_method_diagram.png) ### Concept 3: Special Sequences Examiners expect you to instantly recognise certain famous sequences. Committing these to memory is an easy way to secure marks. - **Square Numbers**: 1, 4, 9, 16, 25... ($n^2$) - **Cube Numbers**: 1, 8, 27, 64, 125... ($n^3$) - **Triangular Numbers**: 1, 3, 6, 10, 15... (Add 2, then 3, then 4...) - **Fibonacci Sequence**: 1, 1, 2, 3, 5, 8... (Add the two previous terms together) ### Concept 4: Linear (Arithmetic) Sequences A linear sequence increases or decreases by the same amount every time. This constant amount is called the "common difference". To find the nth term: 1. Find the common difference (this is the number in front of n). 2. Write out the times table for that number. 3. Find what you need to add or subtract to get to your sequence. ### Concept 5: Quadratic Sequences (Higher Tier) Quadratic sequences have an $n^2$ in their rule. The key feature is that the *first differences* are not equal, but the *second differences* are constant. ![Quadratic Sequences: Second Differences Method](https://xnnrgnazirrqvdgfhvou.supabase.co/storage/v1/object/public/study-guide-assets/guide_dbb772b2-585f-4e65-b9a3-d091bb57070f/quadratic_sequence_diagram.png) To find the nth term: 1. Find the first differences, then the second differences. 2. Halve the second difference. This is the coefficient of $n^2$. 3. Subtract the $n^2$ sequence from your original sequence. 4. Find the linear nth term of the remaining numbers. 5. Combine them. ![Types of Mathematical Sequences](https://xnnrgnazirrqvdgfhvou.supabase.co/storage/v1/object/public/study-guide-assets/guide_dbb772b2-585f-4e65-b9a3-d091bb57070f/sequence_types_diagram.png) ## Mathematical Relationships - **Linear nth term**: $an + b$ (where $a$ is the common difference) - **Quadratic nth term**: $an^2 + bn + c$ (where $2a$ is the second difference) - **Geometric sequences**: $ar^{n-1}$ (where $r$ is the common ratio) ## Listen to the Podcast Review these concepts on the go with our audio guide: ![Audio Guide: Mastering Sequences](https://xnnrgnazirrqvdgfhvou.supabase.co/storage/v1/object/public/study-guide-assets/guide_dbb772b2-585f-4e65-b9a3-d091bb57070f/algebra_sequences_podcast.mp3)

    Key Terms & Definitions

    Sequence
    An ordered list of numbers that follow a specific mathematical rule.
    Term
    A single number or item within a sequence.
    Term-to-term rule
    A rule that describes how to calculate the next term in a sequence using the previous term.
    Position-to-term rule (nth term)
    An algebraic formula that calculates the value of any term based on its position (n) in the sequence.
    Common difference
    The constant amount added or subtracted to get from one term to the next in a linear sequence.
    Fibonacci sequence
    A sequence where each term is the sum of the two preceding terms.

    Worked Examples

    Practice Questions

    Algebra

    Pearson
    GCSE
    Mathematics

    Sequences are the heartbeat of algebraic patterns, testing your ability to spot relationships and express them mathematically. Mastering term-to-term and position-to-term rules is essential for unlocking high marks in your GCSE Mathematics exam.

    4
    Min Read
    3
    Examples
    5
    Questions
    6
    Key Terms
    🎙 Podcast Episode
    Algebra
    0:00-0:00

    Study Notes

    GCSE Mathematics: Algebra - Sequences

    Overview

    Sequences are a fundamental topic in GCSE Mathematics that bridge the gap between simple number patterns and complex algebra. A sequence is simply an ordered list of numbers, but the power lies in finding the rule that governs it. Examiners love sequences because they test multiple skills at once: pattern recognition, algebraic notation, and problem-solving. This topic connects heavily to linear graphs (where the common difference is the gradient) and functions.

    In your exam, you can expect a range of question styles. Foundation tier often focuses on generating terms from a rule or identifying special sequences. Higher tier demands finding the nth term of quadratic sequences or proving whether a specific number belongs to a sequence. Let's break it all down.

    Key Concepts

    Concept 1: Term-to-Term Rules

    A term-to-term rule tells you how to get from one number in the sequence to the very next number. You must know the previous term to find the next one.

    Example: The sequence 3, 7, 11, 15... has a term-to-term rule of "add 4".

    This is simple but limited. If an examiner asks for the 100th term, using a term-to-term rule would take forever! This is why we need position-to-term rules.

    Concept 2: Position-to-Term Rules (The nth Term)

    The "nth term" is an algebraic formula that links the position of a number (n) to its actual value.

    Example: If the nth term is 3n + 2:

    • The 1st term (n=1) is 3(1) + 2 = 5
    • The 10th term (n=10) is 3(10) + 2 = 32

    This is incredibly powerful because it allows you to calculate any term instantly without knowing the previous ones.

    How to find the nth term of a linear sequence

    Concept 3: Special Sequences

    Examiners expect you to instantly recognise certain famous sequences. Committing these to memory is an easy way to secure marks.

    • Square Numbers: 1, 4, 9, 16, 25... (n^2)
    • Cube Numbers: 1, 8, 27, 64, 125... (n^3)
    • Triangular Numbers: 1, 3, 6, 10, 15... (Add 2, then 3, then 4...)
    • Fibonacci Sequence: 1, 1, 2, 3, 5, 8... (Add the two previous terms together)

    Concept 4: Linear (Arithmetic) Sequences

    A linear sequence increases or decreases by the same amount every time. This constant amount is called the "common difference".

    To find the nth term:

    1. Find the common difference (this is the number in front of n).
    2. Write out the times table for that number.
    3. Find what you need to add or subtract to get to your sequence.

    Concept 5: Quadratic Sequences (Higher Tier)

    Quadratic sequences have an n^2 in their rule. The key feature is that the first differences are not equal, but the second differences are constant.

    Quadratic Sequences: Second Differences Method

    To find the nth term:

    1. Find the first differences, then the second differences.
    2. Halve the second difference. This is the coefficient of n^2.
    3. Subtract the n^2 sequence from your original sequence.
    4. Find the linear nth term of the remaining numbers.
    5. Combine them.

    Types of Mathematical Sequences

    Mathematical Relationships

    • Linear nth term: an + b (where a is the common difference)
    • Quadratic nth term: an^2 + bn + c (where 2a is the second difference)
    • Geometric sequences: ar^{n-1} (where r is the common ratio)

    Listen to the Podcast

    Review these concepts on the go with our audio guide:

    Audio Guide: Mastering Sequences

    Visual Resources

    3 diagrams and illustrations

    Types of Mathematical Sequences
    Types of Mathematical Sequences
    How to find the nth term of a linear sequence
    How to find the nth term of a linear sequence
    Quadratic Sequences: Second Differences Method
    Quadratic Sequences: Second Differences Method

    Interactive Diagrams

    2 interactive diagrams to visualise key concepts

    Flowchart for identifying sequence types

    The relationship between position and value

    Worked Examples

    3 detailed examples with solutions and examiner commentary

    Practice Questions

    Test your understanding — click to reveal model answers

    Q1

    Here are the first four terms of a number sequence: 4, 11, 18, 25. Write down the next two terms.

    1 marks
    foundation

    Hint: What is the difference between 4 and 11?

    Q2

    Find an expression, in terms of n, for the nth term of this sequence: 6, 10, 14, 18

    2 marks
    standard

    Hint: Find the common difference first. This is the number that goes in front of n.

    Q3

    The nth term of a sequence is 5n - 2. Work out the 20th term.

    2 marks
    standard

    Hint: Substitute n = 20 into the expression.

    Q4

    A sequence has the nth term n^2 + 3. Is 85 a term in this sequence? Explain your answer.

    3 marks
    challenging

    Hint: Set up an equation: n^2 + 3 = 85 and solve for n.

    Q5

    [Higher Tier] Here are the first five terms of a quadratic sequence: 1, 6, 15, 28, 45. Find an expression, in terms of n, for the nth term.

    4 marks
    challenging

    Hint: Find the first differences, then the second differences. Halve the second difference to find the n^2 coefficient.

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    In this Guide

    Key Terms

    Essential vocabulary to know