---
tags: mth225, dailyprep
---

# Daily Prep 5.6 -- MTH 225

## Overview

This is the first of a two-lesson double-header on a method for solving a particular kind of recurrence relation called a **linear homogeneous recurrence relation**. The method is called the **characteristic root method** and involves changing the recurrence relation into an algebra equation that can be solved using high-school methods, then reassembling the solutions into a formula. In this first part, we'll introduce the method and practice it on some simple cases.

## Learning objectives

**Basic Learning Objectives:** *Before* our class meeting, use the Resources listed below to learn all of the following. You should be reasonably fluent with all of these tasks prior to our meeting; we will field questions on these, but they will not be retaught.

- Identify a recurrence relation as linear, homogeneous, both, or neither.
- State the *order* of a recurrence relation.
- Given a first- or second-order linear homogeneous recurrence relation, write its characteristic equation.
- Given the characteristic equation for a first- or second-order linear homogeneous recurrence relation, solve it using algebra to find the *characteristic roots*.

**Advanced Learning Objectives:** *During and after* our class meeting, we will work on learning the following. Fluency with these is not required prior to class.

- Given the characteristic roots for a first- or second-order linear homogeneous recurrence relation, set up the general solution for the recurrence relation.
- Given the general solution for a first- or second-order linear homogeneous recurrence relation, use the initial conditions to find the particular solution.
- Given a linear, homogeneous, second-order recurrence relation, work through the entire process of constructing a solution for it using the characteristic roots method.

## Resources for learning

**Video:** Watch these videos, which are *not* from the usual playlist but from YouTube. I (Talbert) made these a couple of years ago for MTH 325 (this material used to be in that course, not MTH 225). **They are not captioned** so if you need closed-captioning, you can try YouTube's automatically-generated captions; or just let me know. (Total running time abougt 12 minutes):

<iframe width="560" height="315" src="https://www.youtube.com/embed/4c6Bg2GJvQw" title="YouTube video player" frameborder="0" allow="accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture" allowfullscreen></iframe>

<iframe width="560" height="315" src="https://www.youtube.com/embed/GvOBBcnIjlw" title="YouTube video player" frameborder="0" allow="accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture" allowfullscreen></iframe>

<iframe width="560" height="315" src="https://www.youtube.com/embed/Rwm-NW9Y5iM" title="YouTube video player" frameborder="0" allow="accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture" allowfullscreen></iframe>

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**Text:** This material is covered in [Section 2.4](http://discrete.openmathbooks.org/dmoi3/sec_recurrence.html) starting about halfway through that section.

You are free to search for and use other resources in addition to, or instead of the above, as long as you can work the exercises below.

## Exercises

Once you have watched the videos above, go to this form and complete all the non-optional items on it:

https://docs.google.com/forms/d/e/1FAIpQLSdmNpoQePmvnN13WoY_OgEKJ2hYyLL2a3eBT972eDhXQK1uaA/viewform

## Submission and grading

**Submitting your work:** Your work is submitted when you submit the Google Form. You should receive an email receipt indicating that the work was submitted successfully.

**How this is graded:** The pre-class portion of the Daily Prep is graded either 0 points or 1 point, on the basis of completeness and effort. Wrong answers are not penalized. Earning a "1" requires that you:

- Turn the work in before its deadline;
- Leave no item blank or skipped, even accidentally; and
- Give a good-faith effort at a correct answer on every non-optional item.

More information can be found in the [Specifications for Satisfactory Work in MTH 225](/Cy6P0rGZQzuOM3NwZ3ZuMw) document.

When you arrive for the class meeting, you'll be put into a group of 2-3 to complete a quiz over this material, which will be graded on a 0/1 scale on the basis of correctness.