Recurrence Relationships

Understanding Recurrence Relationships

• Recurrence relationships are equations that give a method for calculating the terms of a sequence, usually in terms of the previous term or terms.
• They involve a sequence of numbers where each term is based on the previous term(s). In other words, the value of a term depends on its predecessors.

Types of Recurrence Relationships

• First order linear recurrence relationships: These involve the current term and one previous term. The formula is usually of the form Un = aUn-1 + b where ‘a’ and ‘b’ are constants.
• Second order linear recurrence relationships: These involve the current term and the two previous terms. The formula is usually of the form Un = aUn-1 + bUn-2 + c where ‘a’, ‘b’, and ‘c’ are constants.

Solving Recurrence Relationships

• To solve a first order linear recurrence relationship, it’s helpful to write out the first few terms to understand the pattern. The general form can then usually be found by examining the differences between the terms.
• Solving a second order linear relationship often involves a little more complexity. In general, the approach involves finding a particular solution that fits the recurrence relationship and then using a complementary function that satisfies the associated homogeneous relationship.

Applications of Recurrence Relationships

• Recurrence relationships come up in a range of real-world applications, from modelling population growth to understanding the behaviour of financial investments.
• For instance, iterative procedures used in algorithms about situation involving repetition or actions dependent on the previous state are often modelled through recurrence relationships.
• Numerical methods for solving equations like Newton-Raphson method also make use of recurrence relationships.

Important Notes

• Be careful of initial conditions when working with recurrence relationships. The sequence can’t start without these.
• Always consider the entire sequence when working out solutions to recurrence relationships. It’s not enough to just know the rule, you need to be able to use it to predict future terms, draw graphs and solve related mathematical problems.

Key Revision Exercises

• Practice the formation and solution of recurrence relationships.
• Ensure you master the differences between first and second order recurrence relationships.
• Study and solve problems from real-world applications like financial modeling, population dynamics etc. This ensures a broader understanding of the concepts.