Daily Prep 5.5 -- MTH 225

--- tags: mth225, dailyprep --- # Daily Prep 5.5 -- MTH 225 ## Overview We've seen how to build recurrence relations and recursive definitions for sequences in all kinds of situations. But recurrence relations aren't often the best way to compute something --- closed formulas are better for that. In this lesson we're going to turn our attention to **solving recurrence relations** --- taking a recursive definition for a sequence and converting it into a closed formula that produces the same terms. We begin by looking at how to tell whether a *proposed* solution to a recurrence relation is or is not actually a solution. ## 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. - Explain what it means to "solve" a recurrence relation and what a "solution" to a recurrence relation is. - Determine if a proposed solution to a recurrence relation fails to be a solution or whether it *might* be a solution based on the terms it generates. **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 a proposed solution to a recurrence relation that appears to be a solution, prove that it really is one. ## Resources for learning **Video:** Watch these videos from [the MTH 225 playlist](https://vimeo.com/showcase/8667148) (total running time 24:10): <div style="padding:56.25% 0 0 0;position:relative;"><iframe src="https://player.vimeo.com/video/641526674?h=c6367f36a1&badge=0&autopause=0&player_id=0&app_id=58479" frameborder="0" allow="autoplay; fullscreen; picture-in-picture" allowfullscreen style="position:absolute;top:0;left:0;width:100%;height:100%;" title="Screencast 5.4: Solutions to recurrence relations"></iframe></div><script src="https://player.vimeo.com/api/player.js"></script> <div style="padding:56.25% 0 0 0;position:relative;"><iframe src="https://player.vimeo.com/video/641541992?h=b352f25862&badge=0&autopause=0&player_id=0&app_id=58479" frameborder="0" allow="autoplay; fullscreen; picture-in-picture" allowfullscreen style="position:absolute;top:0;left:0;width:100%;height:100%;" title="Screencast 5.5: Example of a solution check"></iframe></div><script src="https://player.vimeo.com/api/player.js"></script> <br> **Text:** There's not a single, clear-cut place in the textbook where this material is covered all by itself. However if you go to [Section 2.4](http://discrete.openmathbooks.org/dmoi3/sec_recurrence.html) and read through the examples, there are examples of checking solutions that fit nicely with the videos. 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/1FAIpQLScVbaLHhX0GkO5ZoHlQILuy3fYR-j0eHc80ZSwMoAnZg-78-w/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.