Proof: 2^n is Greater than n^2

[@antariowoods9545]

By far this is the best explanation to this problem I have ever seen. It takes patience but boy is it rewarding to finally understand it. thanks a lot! Subscribed (my final is tomorrow morning)

[@WrathofMath]

Thanks so much, Antario! I always try to craft the clearest explanations I can. I have many other proofs on my channel, induction and otherwise - do let me know if you ever have any specific requests (my lessons this month are Christmas-themed so they have some extra spice). Good luck on the final!

[@jcn268]

No this was poorly explained lmao

[@stoopidoo546]

My university course on mathematical methods for physics (maybe it's called that in english, maybe not) just begun and this was the first problem for us to prove in our homework. I've managed to get toward rewriting 2^(n+1) and solving the binomal on my own but due to never having been confronted with induction before, I quickly ran into troubles afterwards. Your video really helped me grasp this problem (massive thanks), tho I will definitely need a lot more practice to become comfortable with induction.

[@Morejamsthanjimin]

Thank you! I never understood when my prof explained induction, but with your explanation I understood immediately!

[@WrathofMath]

So glad to heat it, thanks for watching!

[@nathanielcabansay4535]

This is surprisingly useful for when you want to know why an algorithm with a time complexity of O(2^n) performs worse than one with a time complexity of O(n²)

[@guilhermemonteiro446]

The best explanation I found on youtube!

[@WrathofMath]

Glad to hear it, thanks for watching!

[@z4_i]

6:12 Why did you go 8 and write 1?

[@WrathofMath]

1) It's valid because +8 is greater than +1 2) We made the change so that we could factor the expression into (k+1)^2

[@benjaminvatovez8823]

Thank you for this video! To people who are less intuitive or less experienced like me, I would suggest to do it this way for any k > 4 : First, 2^(k+1) = 2 . 2^k > 2 . k² and then 2k² > (k+1)² if and only if k²-2k-1 = (k-1)² - 2 > 0, which is true as (k-1)² - 2 > (4-1)² - 2 = 7 > 0

[@ratmouse088]

I agree that this has been the best explanation I've found. Finally get it!

[@mymentalhealthjournal2610]

Where did the 1 come from? Why did you suddenly say that 8 is greater than 1, why is it specifically 1? Or do we let it be specifically 1 because we want to make it become k^2+2k+1?

[@WrathofMath]

Thanks for watching, and exactly! Eight is greater than a lot of things, we only point out the one that's useful!

[@concept-math]

question no. 32. . why we ignored that term. there should be some reason behind

[@dyip-vb1wl]

where did you get the k^2 + 4k

[@Sir_Isaac_Newton_]

Did you watch the video? 2k² = k² + k² > k² + 4k And we know that last "greater than" is true because k > 4

[@Debbs-h]

@@Sir_Isaac_Newton_ where did you exactly come up with the 4k afer having k.k part ?

[@shoopinc]

Thank you, great example of using the information about the integers of interest to run induction.

[@WrathofMath]

Thanks shoop! This is a classic!

[@Mymath.]

Greetings from Turkey, this video's really useful thank you 🙏🏻

[@niha1398]

thank you so much for your efforts my youtube sir

[@xartos299]

Thanks a lot🎉

[@souhir1700]

Thank u so much nice explanation

[@WrathofMath]

Thank you!

[@jcn268]

3:31 is wrong you cant just double up both sides

[@WrathofMath]

It’s correct, you can multiply both sides of an inequality by 2.

[@firstname4337]

this was a great explanation -- but my class wants it done by minimum counterexample (:

[@aleggs2813]

wait so why do we ignore 0 and 1 and jump straight to n = 4? someone please explain. do we just ignore those for some reason?

[@EliCrystal]

because it isn’t true for any numbers less than or equal to 4. for example, with n=3, 2^3>3^2 is False.

[@anty.]

where did the assumption of k*k being greater than 4k come from? like i get it has to do with k being greater than 4 but i don't know what mathematical rule this is. i'm a sophomore taking discrete math and i haven't done any math for over a year lol edit: nvm i get it, since the minimum of k is 5 then that means that k*k is equivalent to at least 5*5 rather than only 4*5

[@sailor5853]

You have k*k, and k > 4 so you can turn one of the k's into a 4 and have k*k > 4*k > 4*4

[@BleniousNgambi]

Thanks alot

[@WrathofMath]

Happy to help!

[@tariqmath1845]

Thnx

[@asaanimals1057]

Thank ❤

[@gani999]

2^n is not greater than n^2 for n = 2 , 3 or 4.

[@WrathofMath]

that's why I specify n>4

[@al8905]

neat

[@user-zy8kr2zy5s]

Help me, What's length and width

[@jakederouin9116]

o K

[@WrathofMath]

Thanks for watching!

[@tapashgayen7237]

This is not tricky it's easy it come in our exam only 2 marks

[@dyip-vb1wl]

Do you want a medal or smth 😂😂