Recurrence relationships
Introduction to Recurrence Relationships
- Recurrence relationships, also known as recursive sequences, are sequences where each term is defined as a function of its preceding term(s).
- A simple recurrence relationship provides a means of generating sequences - however, the first term must be given.
- Recursive sequences are commonly used in a variety of mathematical and real-world scenarios.
Establishing a Recurrence Relationship
- A recurrence relationship is conveyed in this form: u_{n+1}=f(u_n), where u_{n+1} is the next term in the sequence, u_n is the current term, and f(u_n) is a function applied to the current term to find the next term.
- In order to start a sequence, an initial term - typically u_1 - must be stated.
Behaviour of Recurrence Relationships
- The behaviour of the sequence generated by a recurrence relationship can vary based on the function and the initial term.
- A sequence from a recurrence relationship can be finite or infinite, increasing or decreasing, and convergent or divergent.
Examples of Recurrence Relationships
- For example, a simple recurrence relationship is the Fibonacci sequence, expressed as u_{n+2}=u_{n+1}+u_n, where u_1=1 and u_2=1. Each term is the sum of the two preceding terms.
- Standard geometric sequences can also be regarded as examples of recurrence relationships where u_{n+1}=ru_n, and r is the common ratio.
Uses and Applications
- Understanding recurrence relationships can aid in solving complex mathematical problems and find patterns in various types of data.
- They are widely utilised in computer science for programming algorithms, in economics for modelling financial data, and in biology for population growth models.
Revision Tips
- To become proficient in identifying and using recurrence relationships, practice is key.
- It’s beneficial to attempt variety of recurrence relationship problems to understand their different types and behaviours better.
- Continually working with different recurrence relationships will boost confidence and skills ahead of any paper. Always connect the theory with practical examples.