# Lecture 20 – Solving Recurrence Equations

You may want to download the lecture slides that were used for these videos (PDF).

## 1. Review of Recurrence Equations

This video reviews recurrence equations and four examples of where they arise.  Can you determine how fast the solution to a recurrence equation grows, just by looking at the recurrence? (4:37)

## 2. Developing a General Framework (1 & 2)

We consider the family V of all functions which map the set Z of all integers to the set C of complex numbers. V is an infinite dimensional vector space over the field C of complex numbers, with (f + g)(n) = f(n) + g(n) and (α f)(n) = α(f(n)). (6:10)

## 3. Developing a General Framework (3 & 4)

We will first focus on homogeneous linear recurrence equations. (3:25)

## 4. Developing a General Framework (5)

We define the advancement operator A on the vector space V by the rule A f(n) = f(n+1). (2:27)

## 5. Developing a General Framework (6)

This video introduces the following theorem: The set S of all solutions to a homogeneous linear recurrence equation is a d-dimensional subspace of the vector space, V. (3:29)

## 6. The Base Case

This video introduces the following theorem: If r ≠ 0, the 1-dimensional space S of all solutions to the linear homogeneous equation (A – r)f(n) = 0 has the function rn as a basis. Equivalently, all solutions of (A – r)f(n) = 0 are of the form, f(n) = crn for some constant c.  (1:47)

## 7. The Base Case – Proof

In this video, we prove the theorem introduced in the previous video, that if r ≠ 0, the 1-dimensional space S of all solutions to the linear homogeneous equation (A – r)f(n) = 0 has the function rn as a basis. (6:08)

## 8. Towards the General Case (1)

In this video, we verify that the functions (-7)n and 5n are solutions to the equation (A2 +2A-35)f(n)=0.  This is because we can factor the expression, obtaining (A + 7)(A – 5)f(n) = 0. (5:25)

## 9. Towards the General Case (2)

Here we extend our results from the previous slide to the complex case. (2:46)

## 10. Towards the General Case (3)

Here we extend our results from the previous slide to the case with repeated roots. (3:30)

## 11. Towards the General Case (4 & 5)

Here we once again extend our results from the previous slide to the complex case.  Then, we introduce a set of functions that are solutions to the homogeneous equation with a root of multiplicity m. (1:54)

## 12. Towards the General Case (6 & 7)

This video explores two examples of homogeneous equations and their solutions, while drawing connections to linear algebra. (5:33)

## 13. Partial Fractions

In this video, we provide a brief review of partial fractions and make connections between partial fractions and linear algebra. (5:07)

## 14. Differential Equations

This video explores the connections between finite dimensional vector spaces, homgeneous differential equations, and homgeneous recurrance relations. (3:57)

## 15. The Non-Homogeneous Equation

In this video we discuss the significantly more difficult problem of finding a solution to the non-homogenous equation. (2:57)

## 16. Example of a Non-Homogeneous Equation

This video introduces an example of a non-homogenous equation, gives a particular solution, and the general solution. (4:16)

## 17. Another Example of a Non-Homogeneous Equation

This video explores another example of a non-homogenous equation which is more complicated, and discusses some of the mathematics we will explore in the following lecture. (2:56)