Recurrence Relations in Discrete Mathematics: Defining Sequences Recursively

Explore recurrence relations, a method for defining sequences where each term is a function of preceding terms. This guide explains how recurrence relations work, defines the order of a recurrence relation, and provides examples of solving recurrence relations.

Recurrence Relations in Discrete Mathematics

What is a Recurrence Relation?

A recurrence relation is a way of defining a sequence where each term is expressed as a function of previous terms. It's a type of equation that relates a term in a sequence to earlier terms in the same sequence. Recurrence relations are also called difference equations.

Examples of Recurrence Relations

Here are some examples:

f(x + 3h) + 3f(x + 2h) + 6f(x + h) + 9f(x) = 0
a_r = a_r-2 + a_r-1 (Fibonacci sequence)

Order of a Recurrence Relation

The order of a recurrence relation is the difference between the largest and smallest subscripts of the dependent variable. It indicates how many previous terms are needed to calculate the next term.

Examples: Determining the Order

(Two examples showing how to determine the order of a recurrence relation are given in the original text and should be included here.)

Degree of a Difference Equation

The degree of a difference equation is the highest power of the dependent variable in the equation.

Examples: Determining the Degree

(Three examples demonstrating how to find the degree of a difference equation are given in the original text and should be included here.)

Conclusion

Recurrence relations provide a powerful way to define and work with sequences. The order and degree of a recurrence relation are important characteristics that describe the complexity of the relation.