Table Of ContentBrighter Thinking
A Level Further Mathematics
for OCR A
Discrete Student Book (AS/A Level)
Jan Dangerfield
Introduction
You have probably been told that mathematics is very useful, yet it can often seem like a lot of techniques
that just have to be learnt to answer examination questions. You are now getting to the point where you
will start to see how some of these techniques can be applied in solving real problems. However, as well as
seeing how maths can be useful we hope that anyone working through this resource will realise that it can
also be incredibly frustrating, surprising and ultimately beautiful.
The resource is woven around three key themes from the new curriculum.
Proof
Maths is valued because it trains you to think logically and communicate precisely. At a high level, maths is
far less concerned about answers and more about the clear communication of ideas. It is not about being
neat – although that might help! It is about creating a coherent argument that other people can easily
follow but find difficult to refute. Have you ever tried looking at your own work? If you cannot follow it
yourself it is unlikely anybody else will be able to understand it. In maths we communicate using a variety
of means – feel free to use combinations of diagrams, words and algebra to aid your argument. And once
you have attempted a proof, try presenting it to your peers. Look critically (but positively) at some other
people’s attempts. It is only through having your own attempts evaluated and trying to find flaws in other
proofs that you will develop sophisticated mathematical thinking.
Problem solving
Maths is valued because it trains you to look at situations in unusual, creative ways, to persevere and to
evaluate solutions along the way. We have been heavily influenced by a great mathematician and maths
educator George Pólya, who believed that students were not just born with problem-solving skills – they
were developed by seeing problems being solved and reflecting on their solutions before trying similar
problems. You might not realise it but good mathematicians spend most of their time being stuck. You need
to spend some time on problems you can’t do, trying out different possibilities. If after a while you have not
cracked it, then look at the solution and try a similar problem. Don’t be disheartened if you cannot get it
immediately – in fact, the longer you spend puzzling over a problem the more you will learn from the
solution. You might never need to integrate a rational function in future, but we firmly believe that the
problem-solving skills you will develop by trying it can be applied to many other situations.
Modelling
Maths is valued because it helps us solve real-world problems. However, maths describes ideal situations
and the real world is messy! Modelling is about deciding on the important features needed to describe the
essence of a situation and turning that into a mathematical form, then using it to make predictions,
compare to reality and possibly improve the model. In many situations the technical maths is actually the
easy part – especially with modern technology. Deciding which features of reality to include or ignore and
anticipating the consequences of these decisions is the hard part. Yet it is amazing how some fairly drastic
assumptions – such as pretending a car is a single point or that people’s votes are independent – can result
in models that are surprisingly accurate.
More than anything else, this resource is about making links. Links between the different chapters, the
topics covered and the aforementioned themes, links to other subjects and links to the real world. We hope
that you will grow to see maths as one great complex but beautiful web of interlinking ideas.
Maths is about so much more than examinations, but we hope that if you take on board these ideas (and
do plenty of practise!) you will find maths examinations a much more approachable and possibly even
enjoyable experience. Nevertheless, always remember that the result of what you write down in a few
hours by yourself in silence under exam conditions is not the only measure you should consider when
judging your mathematical ability – it is only one variable in a much more complicated mathematical
model!
How to use this resource
Throughout this resource you will notice particular features that are designed to aid your learning. This
section provides a brief overview of these features.
In this chapter you will learn how to:
work with general sequences
use induction to prove results relating to sequences and series
describe behaviour of sequences
use the limits of sequences
work with Fibonacci and Lucas numbers
solve first-order recurrence relations
solve second-order recurrence relations.
Learning objectives
A short summary of the content that you will learn in each chapter.
Before you start…
A Level Mathematics You should be able to use the 1 Given that a sequence has terms
Student Book 2, following formulae for geometric find:
Chapter 4 progressions.
a the term
term of a geometric b the sum of the first five terms.
progression
2 Given that a sequence has terms
find:
a the term
b the sum of the first terms
c the sum to infinity
Pure Core Student You should be able to use the 3 Prove by induction that the sum of
Book 1, Chapter 6 method of induction:
the first integers is .
Check true for initial conditions,
e.g. .
Show true for
(assuming true for ).
Hence, explain true for all
positive integers.
Before you start
Points you should know from your previous learning and questions to check that you’re ready to start the
chapter.
WORKED EXAMPLE
The left-hand side shows you how to set out your working. The right-hand side explains the more
difficult steps and helps you understand why a particular method was chosen.
PROOF
Step-by-step walkthroughs of standard proofs and methods of proof.
Key point
A summary of the most important methods, facts and formulae.
Explore
Ideas for activities and investigations to extend your understanding of the topic.
Tip
Useful guidance, including on ways of calculating or checking and use of technology.
Each chapter ends with a Checklist of learning and understanding and a Mixed practice exercise,
which includes past paper questions marked with the icon .
In between chapters, you will find extra sections that bring together topics in a more synoptic way.
CROSS-TOPIC REVIEW
EXERCISE
Questions covering topics from across the preceding chapters, testing your ability to apply what
you have learnt.
You will find practice paper questions towards the end of the resource, as well as a glossary of key
terms (picked out in colour within the chapters), and answers.
Maths is all about making links, which is why throughout this resource you will find signposts emphasising
connections between different topics, applications and suggestions for further research.
Rewind
Reminders of where to find useful information from earlier in your study.
Fast forward
Links to topics that you might cover in greater detail later in your study.
Did you know?
Interesting or historical information and links with other subjects to improve your awareness
about how mathematics contributes to society.
Colour-coding of exercises
The questions in the exercises are designed to provide careful progression, ranging from basic fluency to
practice questions. They are uniquely colour-coded, as shown here.
1 Find the recurrence relation for each of these sequences.
a
b
c
d
5 The Fibonacci sequence is defined by .
a Find the first eight terms of the sequence.
b Find the first seven terms of the sequence obtained by taking the differences between
consecutive terms of the Fibonacci sequence.
c Write down the first seven terms of the sequence obtained by taking quotients of consecutive
terms of the Fibonacci sequence (as decimals to decimal places).
d A rectangle has area square unit. Its length is and its width is .
i Form a quadratic equation in and solve it to find the value of .
ii Write down the name of this number.
7 In 2017 the population of the United Kingdom is taken to be million people. The annual growth
rate is . In addition, it is expected that the net immigration will be people per year.
Set up a recurrence system to model the population and solve it to estimate the number of people
in years’ time. Show that the population of the United Kingdom in 2022 will be million
people.
13 A sequence is defined by , with .
Prove by induction that for all positive integers .
Verify that your proof is valid for the first five terms of the sequence.
15 A sequence is defined by , with .
Prove by induction that for all positive integers .
Verify that your proof is valid for the first five terms of the sequence.
Black – drill questions. These come in several parts, each with subparts i and ii. You only need attempt
subpart i at first; subpart ii is essentially the same question, which you can use for further practice if
you got part i wrong, for homework, or when you revisit the exercise during revision.
Green – practice questions at a basic level.
Yellow – designed to encourage reflection and discussion.
Blue – practice questions at an intermediate level.
Red – practice questions at an advanced level.
– indicates content that is for A Level students only.
– indicates content that is for AS Level students only.
Contents
Introduction
How to use this resource
1 Mathematical preliminaries
Section 1: Classifying objects
Section 2: Counting methods
Mixed practice 1
2 Graphs and networks
Section 1: The language of graph theory
Section 2: Types of graph
Section 3: Using graphs
Mixed practice 2
3 Algorithms
Section 1: Understanding algorithms
Section 2: Comparing algorithms
Section 3: Sorting and packing problems
Mixed practice 3
4 Network algorithms
Section 1: Least weight path between two vertices
Section 2: Least weight tree connecting all the vertices
Section 3: Least weight cycle through all the vertices
Section 4: Least weight route using every arc
Mixed practice 4
Cross-topic review exercise 1
5 Decision making in project management
Section 1: Critical path analysis
Section 2: Scheduling
Mixed practice 5
6 Graphical linear programming
Section 1: Formulating linear programming problems
Section 2: Working with constraints
Section 3: Graphical solutions
Mixed practice 6
7 The simplex algorithm
Section 1: Carrying out the simplex algorithm
Section 2: Interpretation
Mixed practice 7
8 Game theory
Section 1: Simple two-person, simultaneous play games
Section 2: Mixed strategies
Mixed practice 8
Cross-topic review exercise 2
Practice paper
Formulae
Answers
1 Mathematical preliminaries
2 Graphs and networks
3 Algorithms
4 Network algorithms
5 Decision making in project management
6 Graphical linear programming
7 The simplex algorithm
8 Game theory
Glossary
Acknowledgements
Copyright
1 Mathematical preliminaries
In this chapter you will learn how to:
classify four types of problem
partition a set
use the pigeonhole principle
enumerate arrangements
enumerate selections
use the inclusion–exclusion principle.
Before you start…
1 A cafeteria has a choice of three types of drink and four types of
GCSE You should be able to
cake. In how many different ways can a customer choose a drink
use systematic listing
and a cake?
strategies.
2 Draw a Venn diagram to show the set of positive integers that are
GCSE You should be able to
less than and the following subsets.
use Venn diagrams and
sets. a is odd
b is a square number
