AEP 5: Polynomial formulas for sequences

--- tags: mth225, 225-spr22, aep --- # AEP 5: Polynomial formulas for sequences ## Overview We've seen ways to construct closed formulas for sequences in special circumstances. If the sequence is arithmetic or geometric, closed formulas are fairly simple to build; and we can use the method of characteristic roots if the sequence has a linear homogeneous recurrence relation. In this AEP, you'll be working through a section of the text that covers another method, which applies when the differences in the terms of the sequences obey a certain pattern. When that pattern holds, we can find a *polynomial function* for the sequence's closed formula. ## Background The background material for this AEP is found in [Section 2.3 of the Levin text](http://discrete.openmathbooks.org/dmoi3/sec_polyfit.html). Before beginning the tasks below, you'll need to read through that section and work through the examples. There are exercises at the end as well that can give you further practice and will make this AEP go faster for you. ## Tasks for this AEP 1. Consider the sequence $1, 3, 7, 13, 21, \dots$. Is this sequence $\Delta^k$-constant for some value of $k$? (That term is the central focus of the reading. You did the reading, right?) If so, what's the value of $k$? Be sure to explain your reasoning. 2. Repeat for the sequence $2, 11, 34, 77, 146, 247,\ldots$ 3. Repeat for the sequence $4, 12, 36, 108, 324, \dots$ 4. Use the techniques of Section 2.3 to find a polynomial closed formula for the sequence in the first problem. 5. Repeat for the sequence in the second problem. 6. Work all of Exercise 12, about pirates stacking cannonballs (!), from the Levin text section 2.3. ## Expectations and Grading Criteria AEPs are graded using the "EMRN" rubric found in the syllabus. Make sure you review the [Standards for Assessments in MTH 225](/KoT83ezHRYO3DqPyXMMMag) document before you submit any work, so you're fully aware of the expectations for the different marks. In particular: - All work needs to be shown *and* your thought processes clearly expressed in all of the tasks of the assignment. The results also need to be correct. You are not just doing math; you are explaining things to a reader, so a mix of math and English is needed. - All the information needed for an "outsider" to understand your work needs to be self-contained within the work. **The reader should not have to do any work to fill in gaps.** Please note, it is the case with all AEP's that **your grade is primarily based on your explanations and writing, and only secondarily on the precision and correctness of your computations.** Correct computations with insufficient explanation will need to be revised and may receive an "N" grade. A grade of "E" is given if all of the above expectations are met, and the work is of superior quality in terms of the clarity of explanations and work, appearance of the writeup, and precision of the mathematics. ## Submitting your work **AEP submissions must be typewritten and saved as either a PDF or MS Word file. No part of your submission may involve handwriting; work that is submitted that contains handwriting will be graded N and returned without feedback.** This includes electronic handwritten docments, for example using a stylus and a note-taking app. To type up your work, you can use MS Word or Google Docs (both of which have equation editors for mathematical notation) or any other computer-based math typesetting tool. Just make sure you save your work as a Word document or PDF (no `.odt`, `.rtf`, or other file extensions are allowed). When you have your work typed up, double-check it for neatness, correctness, and clarity. Then simply submit your document on Blackboard, in the **AEP** area, in the **AEP 5** assignment. ## Getting Help You **may** ask me (Talbert) for help on this assignment in the form of **specific mathematical or technical questions, or clarifying questions about the instructions**. If I cannot answer a question because it would give too much away, I'll tell you so. However please note: **I will not "look over your work" before you submit it to give you feedback on the overall submission**. I have made the expectations clear, so just follow those directions and submit your best work, and you'll be allowed to revise it if needed. For AEPs, the syllabus policy on collaboration is: >**No collaboration is allowed at all** — with other people, or with print or electronic sources other than your textbook, the video playlist, or your notes. **You can ask technology related questions to anyone at any time**. For example if you need help figuring out how to type up your work, there are no restrictions on that.