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Introduction to Generating Functions for A-Level Maths Students

Learn the fundamentals of generating functions and experience studying maths at a university level.

607 enrolled on this course

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  • Duration

    3 weeks
  • Weekly study

    3 hours

Discover what it’s like to study mathematics at University

This three-week course from Manchester Grammar School will give you an opportunity to study a university-level topic in-depth.

As an A-level mathematics student, you’ll be able to further your learning through the study of generating functions and obtain a sense of what it’s like to study maths at university.

Explore the capabilities of generating functions as a problem-solving tool

Generating functions is a powerful mathematical tool that can be applied to solve problems in many different contexts.

On this course, you’ll be introduced to generating functions and its applications in mathematics. In the first week of the course, you’ll define generating functions and work with various mathematical problems to understand the types of problems generating functions can solve.

Learn how to use generating functions for probability distributions and more

This course will cover roughly one topic per week and is structured to provide you with an in-depth understanding of how to build generating functions for a wide range of mathematical problems.

You’ll learn how to derive the nth term for a selection of recurrence relations, solve combinatorial problems, work with standard probability distributions, and more, all using generating functions.

Decide your academic trajectory with help from an expert educator

The lead educator on this course is an experienced A-Level teacher who has studied generating functions at a high level during his own undergraduate and postgraduate studies.

With the insight and expertise of your lead educator, you will have an understanding of what it means to study maths at a higher level, and be better equipped to make a decision in pursuing maths at university.


  • Week 1


    • Introduction to the course

      I will introduce the course and give you some mathematical problems to try. These problems will be referred to later in the course and will give you an idea of the types of problems generating functions can help us to solve.

    • Introduction to generating functions

      In this activity, generating functions will be formally defined. This is the most important lesson of the entire course because all future activities will build upon the ideas introduced here.

    • Recurrence relations

      In this activity we will consider how generating functions can be used to solve recurrence relations.

  • Week 2

    Combinatorics and generating functions

    • Combinatorics overview

      A brief recap of the basics of combinatorics

    • Building generating functions

      In this activity we will learn how to build generating functions that help us to solve combinatorics problems.

    • Solving combinatorics problems

      In this activity we will learn how to solve a variety of combinatorics problems using generating functions.

  • Week 3

    Probability and generating functions

    • Probability generating functions

      In this activity we will learn how to build generating functions for probability distributions.

    • Sums of random variables

      In this activity we will consider an important result which enables us to quickly find the generating function for the sum of random variables.

    • Random walks (advanced)

      In this activity we will introduce a process known as random walks and a related generating function.

    • Closing remarks

      A final chance for you to share your thoughts with other learners and closing remarks from the course creator.

Learning on this course

On every step of the course you can meet other learners, share your ideas and join in with active discussions in the comments.

What will you achieve?

By the end of the course, you‘ll be able to...

  • Explain what generating functions are and why they are useful
  • Apply generating functions to find the nth term for sequences defined using recurrence relations
  • Solve combinatorics problems
  • Apply generating functions in the context of combinatorics
  • Apply generating functions in the context of probability distributions

Who is the course for?

This course is designed for A-Level mathematics students who are considering applying to study mathematics at university, but anyone with an interest in mathematics is welcome to join the course.

What software or tools do you need?

We will be using Wolfram Alpha in the course to do simple mathematical calculations (the free version is absolutely fine).

Who will you learn with?

Mathematics Teacher at The Manchester Grammar School with a particular interest in Mechanics and Probability. I am always on the lookout for interesting ways to introduce topics in the classroom.

Who developed the course?

Manchester Grammar School

Our history dates back to the time of Henry VIII, when The Manchester Grammar School was founded in 1515 by Hugh Oldham, Bishop of Exeter, to provide ‘godliness and good learning’ to the poor boys of Manchester.

The School proceeded to build a reputation as one of the country’s leading educational establishments, a position it still holds today as an independent day school.

Learning on FutureLearn

Your learning, your rules

  • Courses are split into weeks, activities, and steps to help you keep track of your learning
  • Learn through a mix of bite-sized videos, long- and short-form articles, audio, and practical activities
  • Stay motivated by using the Progress page to keep track of your step completion and assessment scores

Join a global classroom

  • Experience the power of social learning, and get inspired by an international network of learners
  • Share ideas with your peers and course educators on every step of the course
  • Join the conversation by reading, @ing, liking, bookmarking, and replying to comments from others

Map your progress

  • As you work through the course, use notifications and the Progress page to guide your learning
  • Whenever you’re ready, mark each step as complete, you’re in control
  • Complete 90% of course steps and all of the assessments to earn your certificate

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