Prove that n^3 + 2n is divisible by 3 using Mathematical Induction
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- เผยแพร่เมื่อ 26 เม.ย. 2022
- Prove that n^3 + 2n is divisible by 3 using Mathematical Induction
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Thank you:)
I love induction proofs, both the weak and strong forms. So, do some more of these The Z is Zahlen in German. The early mathematicians who developed set theory (like Cantor) used this letter. N for natural numbers,R for real numbers, Q for the rationals. Your proof was exceptionally well done. Two thumbs up
Recently did a similar proof over on my channel, except I used something similar to the pigeonhole principle. It's the "Indonesian Olympiad" question in my play list. Since we want that n^3 + 2n ≡ 0 (mod 3) and we can factor n^3 + 2n = n(n^2 + 2), we can consider the following. If n ≡ 0 (mod 3), then obviously the product is divisible by 3. If n ≡ 1 (mod 3), then (n^2 + 2) = 9k^2 + 6k + 3, which is clearly divisible by 3. Last if n ≡ 2 (mod 3), then we can write (n^2 + 2) as (9k^2 - 6k + 3) which again, is divisible by 3.
This seems really involved for what it is. It’s a nice introduction to induction, but really modular arithmetic makes this ludicrously short with three cases to check.
Doing induction in school right now, had this exact problem and didn't know how to do it and you upload this lol
Nice hehe
I would love to see a video that misuses mathematical induction. For example, proving that all horses are the same color.
Fascinating! Keeping all the steps allows us to follow along with this beautiful mathematical proof.
Thanks
Wow, that's a great video! Really, I think thanks to that I fully grasped the concept of mathematical induction proof :))
Great Induction Proof! I like how you handled both the inductive hypothesis and the inductive step. Using the inductive hypothesis is often difficult for students learning induction for the first time.
By the way, The Principal of Mathematical Induction is a school administrator. I think you meant the theorem, Principle of Mathematical Induction.
Omg I spelled it wrong !
Thx😀
I solved it the same way on my own! Have you ever demonstrated any strong induction problems? I don't think I ever understood that process quite right.
I should make some videos on that!
Beautiful
What is the background tool you are using?
it's an older program called smoothdraw
@@TheMathSorcerer Thank you!Nice proof!
top notch video brother
thanks man
I don't understand if ,by your proof, this propriety is valid for n ∈ N, or also for n ∈ Z. That's because, from what is my grasp on induction, if u want to prove it also for negative numbers u should show that if the propriety is valid for n, then is valid for n-1. Am i wrong?
Since n^3 + 2n is an odd function, replacing n with -n, you get -(n^3+2n), and since n^3+2n is divisible by 3 so is its negative