MTH 225: Weekly Challege 10

--- tags: mth225, weekly-practice --- # MTH 225: Weekly Challege 10 :::danger **Weekly Challenges 8, 9, and 10 are deadline-free.** This means that you may submit your work whenever you believe it is ready to be assessed. However, remember there is one final deadline: *11:59pm Eastern on Sunday, December 12* which is the final deadline for all submissions and revisions of Weekly Challenge work. No submissions or revisions of Weekly Challenges will be accepted past this point. In order to give yourself enough time and space to revise your work on Weekly Challenges 8, 9, and 10, make every effort to submit a completed draft as soon as possible. ::: :::warning **Reminders**: * Your work on Weekly Challenges should consist of **complete solution attempts for all the Application/Extension Problems and complete and thoughtful responses to all the Feedback and Reflection prompts**. Before submitting your work, make sure you've reviewed the [Specifications for Satisfactory Work in MTH 225](/Cy6P0rGZQzuOM3NwZ3ZuMw) document to make sure your work meets the standards to the best of your knowledge. * You may type up your work or write it by hand on paper, whiteboard, or in a notes app. **Typewritten work is preferred** because it makes revisions easier for you. * If you handwrite your work on paper or a whiteboard, your work needs to be **scanned to a legible, black-and-white PDF**. * All your work is to be submitted as a **single PDF** at the appropriate assignment area on Blackboard in the *Weekly Challenges* folder. Please do not submit multiple PDFs, or files that are not in PDF format. ::: --- *There are no practice problems or reflection questions this time.* :::info In this Weekly Challenge, there are four statements that can be proven using mathematical induction. You can complete this Weekly Challenge by choosing one of two options: **Option 1:** Choose **one** (1) of these statements and give a complete, clearly-stated framework for the proof, **then write out a completed proof for it.** If you choose this option, you'll need to do the following: - Write up the framework and the proof in different sections of your writeup, to keep them separated. - The framework needs to follow the specifications stated below in Option 2 for frameworks. - Your proof needs to be clearly written, mathematically correct, and neatly written up (excessively messy or disorganized writing will need to be revised even if the math and logic are correct!), and your proof must follow the framework you set up. (I.e. you should just be filling in the details of your framework and then making it all look nice in a narrative format.) **Option 2:** Choose **two** (2) of the statements and **give a complete, clearly-stated framework of an induction proof for each statement you chose**. A completed proof is not necessary; just two well-constructed frameworks. Satisfactory work on a framework for an induction proof means: - The base case is correct, clearly stated, and proven. (In other words you *do* need to actually prove the base case.) - The induction hypothesis is clearly stated in specific terms, using the predicate in the statement. It should begin with "Assume that..." and then state clearly and specifically what needs to be assumed. - The framework clearly states what needs to be proven once the induction hypothesis is stated. This should begin with "We want to prove that..." and then state clearly and specifically what needs to be proven next. Then, to finish, give an outline of how the proof might proceed. Again you do *not* need to give a completed proof (if you think you can do a completed proof, you should consider Option 1!) but you do need to state in specific terms what you might do in a completed proof. For examples of good frameworks and completed proofs, you can consult the Class Page, the Jamboards from Week 14, and your textbook. :::success Remember that **a Satisfactory mark on this Weekly Challenge will be considered as a successful demonstration of skill on Learning Target RI.6, which will give you "fluency" on that target**. Please see [this announcment on Campuswire](https://campuswire.com/c/GB1A69E25/feed/127) for more. ::: Again, choose ONE (1) of the statements below if you are doing Option 1 (framework + completed proof), and TWO (2) of these if you are doing Option 2 (two frameworks). Submissions that have work on more than two of these will be marked Incomplete and returned without comment. **Please clearly state which option you are choosing** so I'll know how to grade it. ### Statement 1 For every integer greater than $6$, $3^n < n!$. ### Statement 2 For every natural number $n$, $\binom{n}{0} + \binom{n}{1} + \binom{n}{2} + \dots + \binom{n}{n} = 2^n$. ### Statement 3 Suppose $n$ is a positive integer and $A_1, A_2, A_3, \dots, A_n$ and $B$ are sets. Then $$(A_1 \cap A_2 \cap A_3 \cap \dots \cap A_n) \cup B = (A_1 \cup B) \cap (A_2 \cup B) \cap (A_3 \cup B) \cap \dots \cap (A_n \cup B)$$ ### Statement 4 Denote the $i$th Fibonacci number by $F_i$ with $i = 1, 2, ...$ so that $F_1 = 1, F_2 =1, F_3 = 2, F_4 = 3, F_5 = 5, F_6 = 8,$ and so on. Remember these are defined by the recurrence relation $F_1 = 1, F_2 = 1$ and for $n > 2$ we have $F_n = F_{n-1} + F_{n-2}$. Claim: For all integers $n \geq 2$, $F_{2n} = \left( F_{n+1} \right)^2 - \left( F_{n-1} \right)^2$