Math 194: Problem Set #2
Spring 2024
Problem 1
Read the attached introduction to mathematical induction. Then grade the text based on the attached grading rubric.
Problem 2
Read the attached essay “How to write a paper” by Oded Goldreich. Then pick a target audience for reading an introduction to mathematical induction (e.g. college students in an introduction to proofs class, advanced high school students, program-mers working at Google, or ....) and rewrite the attached introduction to mathemat-ical induction as a well-written introduction to that audience. Your text should be typed.
Collaboration Policy
With each week’s homework, you must turn in a one paragraph description of all the resources you used on that homework. You must mention any person you talked to about the problems, any book you looked at, any online resource (Wikipedia, Chegg,...) that you used. A sample paragraph is
On this week’s homework, I worked on the problem set collaboratively with Gauss and Grothendieck at The Redroom during happy hour. We found an Alex Jones video (http://youtube.blah.com) that gave a really clear explanation of Fermat’s Last Theorem. We compared our solutions against a solution key that we found on the /commutativealgebra/ board of 4chan (http://blah.blah.edu). We also got really stuck on Problem 5, and so we went to Chegg.com and paid an online tutor (“Zariski”) $50 to solve the problem for us.
An Introduction to Mathematical Induction:
Mathematical induction is a great tool in our mathematical toolbox, a bit like a magic wand that helps us prove things about numbers. It’s a little tricky at irst, but once you get the hang of it, you’ll be rocking it like a wizard in no time.
Suppose we want to prove something for every number in a sequence, but there are ininite numbers and we are lazy. That’s where induction comes in handy.
Step 1: Base Case
First, we must establish that everything we are trying to prove is valid for theirst number in the sequence. We call this the ”base case” because it is like the bottom of a mathematical skyscraper. Without it, everything falls apart.
Step 2: Inductive Step
Once we have our basic scenario out of the way, let’s move on to the fun part: the induction phase. This is where we get a little more stealthy. We assume that everything we are trying to prove is true for a number, let’s call it ”n”. Then we show that if it is true for “n,” then it must also be true for the next number, “n + 1.”
It’s a bit like a relay race. When the baton (or in this case our mathematical truth) passes from one runner to another without dropping it, we know we are on the right path.
Step 3: Repeat Ad Ininitum
We keep on repeating this process, showing that if our statement holds true for any given number, it must also hold true for the next one. And since there’s no end to the numbers (they just keep going and going like the Energizer Bunny), we can be pretty conident that our statement holds true for all of them.
On this page, the process must be kept going, showing our statement holding true for any certain number means also that it holds for the next one. And since the numbers have no end (they just go on and on like the Energizer bunny), we can be very conident that what we say applies to everyone.
Conclusion
By combining the base case and the trigger step, we can check that our statement is true for any number in the sequence. It’s like magic—but with numbers!
So, next time you’re faced with a daunting task of proving something about a whole bunch of numbers, remember the power of mathematical induction. It might just save the day.
Application: Proving the Sum of Squares
Alright, bros, here’s where things get real interesting. Let’s say we wanna prove that the sum of the squares of the irst ”n” natural numbers is given by the formula:
. We start of with our base case, showing that the formula holds true for the irst natural number. Then, we hit up the inductive step, assuming it’s true for ”k” and proving it for ”k + 1”. By following the steps of induction, we can conidently slam dunk this proof and show that the formula holds true for all natural numbers ”n” .
In Conclusion
By crushing the base case, keeping the party going with the inductive step, and throwing in some dope applications like the sum of squares, we’ve mastered the art of mathematical induction. It’s like being the MVP of the math game – unstoppable, bro!
So, next time you’re faced with a math problem that seems impossible, just re- member the power of mathematical induction. It’s like the secret sauce that makes every party epic!