Sequences & Recursion

Sequences are ordered lists of objects that follow specific patterns or rules. Each number in the sequence is called a term. Sequences can be classified as arithmetic or geometric. Arithmetic sequences may be used for modelling linear patterns, while geometric sequences can model more complex numerical patterns.
Sequences are important in mathematics because they are the foundation for many other mathematical concepts and they have many applications in other fields. Sequences are used to model financial scenarios, computer algorithms, signal processing in engineering and in the study of genetics in Biology. These applications show how sequences are not just abstract mathematical concepts but are integral to solving real-world problems and advancing various scientific fields.
In a recurrence relation, the terms of a sequence are dependent on the previous terms of the sequence. A first- order linear recurrence relation is a relation whereby the terms of the sequence depend only on the previous term of the sequence, which means that we need only an initial value to be able to generate all remaining terms of the sequence.

In a recurrence relation, the nth term is represented by \(u_n\), with the term directly after \(u_n\) being \(u_{n+1}\) and the term before \(u_n\) being represented by \(u_{n-1}\). The initial value of the sequence is represented by the term \(u_0\).

If the initial value in a recurrence relation changes, then the whole sequence changes. If we are not given an initial value, we cannot determine any terms in the sequence.

This resource revises sequences and recursion and within sequences and recursion there are 4 key concepts:

• Arithmetic sequences
• Geometric sequences
• Recurrence relations
• Recursion for financial mathematics

To determine if this resource will benefit you, start by answering the following questions.

• What is the difference between an arithmetic and geometric sequence?
• How is the equation for an arithmetic sequence determined?
• How does the common ratio influence the behaviour of a geometric sequence?

The answers to these questions are provided on the following pages. Use this resource to refresh your memory, reinforce your understanding of these concepts, and prepare more effectively for university-level learning.

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