Module title & Course code
Numbers and Groups
MAS114
Lecturer
Dr Eugenia Cheng
Course description
The module provides an introduction to more specialised Pure Mathematics. The first half of the
module will consider techniques of proof, and these will be demonstrated within the study of
properties of integers and real numbers. The second semester will study symmetries of objects, and
develop a theory of symmetries which leads to the more abstract study of groups.
University course content
A Level course content
Module
Functions
Domain and range of functions.
Composition of functions.
Inverse functions – including effect on domain and range and which types of
mapping have inverses.
Sketching graphs of functions involving the modulus symbol – in particular
students need to sketch 𝑓|𝑥|𝑎𝑛𝑑|𝑓(𝑥)| given f(x)
Solving equations involving the modulus symbol
C3
Further
module
only?
No
Induction
Proving sums of series e.g. ∑𝑛𝑟=1 𝑟 , ∑𝑛𝑟=1 𝑟(𝑟 + 1)
Proving divisibility e.g. 𝑛3 − 𝑛 is divisible by 6
Proving the nth term of a sequence defined by a recurrence relation, e.g. 𝑈1 =
2𝑈 −1
2
1, 𝑈𝑛+1 = 𝑛3 , prove that 𝑈𝑛 = 3(3)𝑛 − 1
1 0
Proving formulas for powers of matrices, e.g. 𝐴 = (
) , prove that 𝐴𝑛 =
−1 2
1
0
(
)
1 − 2𝑛 2𝑛
FP1
Yes
Proof
Although many proofs may be shown by teachers, most are of their own
discretion. The only examinable proofs are;
𝑛
The proof that for an arithmetic sequence 𝑆𝑛 = 2 (2𝑎 + (𝑛 − 1)𝑑)
C1
No
𝑎(1−𝑟 𝑛 )
The proof that for a geometric sequence 𝑆𝑛 = 1−𝑟
The proof of de Moivre’s theorem and it’s applications e.g. given that
1
𝑧 = cos 𝜃 + 𝑖 sin 𝜃, prove that 𝑧 𝑛 + 𝑛 = 2 cos 𝑛𝜃
𝑧
C2
No
FP2
Yes