---
tags: mth225, dailyprep
---

# Daily Prep 5.7 -- MTH 225

## Overview

In this lesson, we introduce a modification of the characteristic root technique for solving second-order linear homogeneous recurrence relations, that's needed when there's only one characteristic root instead of two.

## 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 the *multiplicity* of a root of a quadratic.
- Determine when a second-order linear homogeneous recurrence relation has a repeated root.
- Set up the modified framework for a second-order linear homogeneous recurrence relation with a repeated root.

**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.

- Go through the complete solution process for a second-order linear homogeneous recurrence relation with a repeated root, and check the solution.

## Resources for learning

**Video:** Watch this from [the MTH 225 playlist](https://vimeo.com/showcase/8667148) (total running time 11:18):

<iframe src="https://player.vimeo.com/video/646042713?h=b0d253b665" width="640" height="360" frameborder="0" allow="autoplay; fullscreen; picture-in-picture" allowfullscreen></iframe>

**Text:** This is at the end of [Section 2.4 in your text](http://discrete.openmathbooks.org/dmoi3/sec_recurrence.html), starting just before Example 2.4.7. That example is a good one to study.

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/1FAIpQLSf1Y_HDnboc64drToj8zUsF8IPitC0Fh73AsSE9F7qME7exfg/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.