I love this channel. Im an aspiring mathematician and frequently encounter overwhelming self-doubt about my ability. But when you explain something and reassure the audience that you struggled also, it is uplifting to know that it is not just me struggling with seemingly easy concepts. Seriously, thank you so much for this.
You amazed me. I just came from Eddie woo and others for this question now YOU! It’s like you understand the most basic intuition needed to solve it and you did it in so little steps. your solution is gold man. even for the factorial question. THANK YOU SO MUCH
This is an extremely good video because you stumbled (or pretended to :) ) a couple times and talked us through how you figured it out. That’s super helpful. Thank you.
I've been struggling a great deal in my proofs class and was self-conscious about my ability to think critically because of it. After watching this, not only do I understand the concept, I feel that I have a greater understanding of how a proof proves its claim. Thank you so much for this video, it has helped immensely!!!
Thx, this is a hard topic to explain! I remember learning this myself and just not getting it. I ended up giving up and only understood it a year later when I looked at it again.
My teacher tried to prove instead that the difference between the inequalities is bigger than zero, I myself find that much more confusing so when I saw this, I was able to solve any problem of inequalities, thanks alot you are going to save my grades.
Thank you so much! You really inspire to continue on with school through this math stuff. Sometimes I feel very unmotivated with math because I'll try and I'll try, and when I get it, it's awesome. Plus it's something I genuinely enjoy, so it sucks sometimes when something is just not clicking. Anyhow, I've been watching some of your videos apart from the instructional math ones and they're definitely inspirational, thanks!
I appreciate the intuitive approach you take - so much of PMI instruction involves chaotic jumps in reasoning that are hard for listeners to follow and seemingly impossible to intuit ("how did you know to do that?"), so your decision to work with a problem you didn't already know is a great help. :) I was able to get this one a different way, but I had to use a pretty ugly derivative in the middle; your method is much more elegant.
For clarification, I know I am very late to responding to this video, however, when you use k=4 you must be sure that the inductive hypothesis hold for that value of k. If you plug 4 into inductive hyp it actually fails to be true. You must use a value for k that you know the inductive hyp holds true for. In this case it would need to be k=5.
What are you talking about its an inequality he didnt 'plug in k=4' he replaced it! I e he threw it in the bin and replaced it with something we know for a fact is smaller than k . Since k is bigger than 4 replacing k with 4 forces an inequality it is him reshaping it so it ends up looking like the conclusion.
That was so cool. I am barely starting my classes for my degree and I understood nothing, but it was very cool seeing you work out the problem. Some day I’ll get it.
Thank you. I was stuck on the '2^1 x 2^k' for a really, really long time. Induction is tough, and I am really overwhelmed but this video has helped me feel better.
Aww man, this was beautiful, you were down to earth and showed very clearly all the things I missed from too many conversations with my professor. I actually have a good idea now, of how I should think when doing these inequality proofs. Absolutely amazing. Thank you!!
2:50 /3:20 /4:47 When dinosaurs roamed the planet xDDDD I love the humility. These are starting to click for me and it's exciting to mess with algebra like this
Best explanation I heard. First I thought this problem and my assignment from my pre-cal class was the same but it was actually the opposite " Prove n^2 > 2^n for n >= 5 " After watching the vid, I knew that the statement is already false so how do I show that the statement above is false using The Mathematical Induction?
Think I'm a bit clear now on how to approach and tackle questions which involve proving inequalities using induction. Thanks a lot for making this video!
A saving grace for discrete math this semester :)) Sets theory proofs and now I found out you do induction too, LETS GOOO!!! I was wondering since we were working with k > 4 how you were able to substitute k = 4 into the equation. Because of the I.H it is totally plausible to do this but it would have to be k >= 4. Even for k>=4 this should work right? I assumed since k > 4 that we were only allowed to plug in 5 or greater for k since our I.H is greater than 4 not equal to it. Thank you!
I think because for the rule k>4, if we substitute k=4 in the LHS equation, then we know the LHS will be bigger than the substituted version of it because of the rule k>4. I think you can also you k=5, but then you have to use >= sign, since LHS can be bigger or equal to k=5 substituted version of it
First of all, thanks for the video everything is more clear now. Today I had my first exam at college and I had to proof that if A is a countable set then so is A^n by induction. Can you make a video of that?
great video!! thanks for sharing your knowledge. I have a question related to the substitution done in the minute 5:00 of the video. You said that "..you allow to do that (the substitution of 2^k by k^2) in math" and change the '=' symbol by '>'. I really want to understand how this substitution is possible and I want to know if you could provide us with any reference or material in which we could go deeper into this subject. Thanks in advance and again, thanks for sharing.
Its because he is replacing 2^k with something he knows is smaller than it namely k^2 so obviously the equality does not hold anymore so he must write the greater than symbol.
@@DodiHD I don't think he messed up. He is not saying k=4, the inequality says > k^2+kk, so whatever is on the left side is greater than this. So using 4, we are saying it will be greater than the value obtained when substituting 4.
I took discrete math 1 year ago. I didn't understand mathematical induction. This semester I am taking theoretical CS and mathematical induction is needed so I am learning it again. This is the first time I understood a proof by Mathematical inducton.
How I did: Checking base case is easy... I proved another inequality before that: 2^m>2m+1 (for m>4) Make hypothesis and other stuff... To proof: 2^m+2^m>2(m+1)+1 (m>4) This reduces (by hypothesis) 2^m>2 (m>4) Works! Nice! Now to the main thing: Do hypothesis and base checking... To proof 2^n+2^n>(n+1)²=n²+2n+1 This reduces to(by hypothesis): 2^n>2n+1 Proved above! So, hence proved. I suppose. Is that right? I wrote it informally... Would do better in exam... I should have gone the other way round like first write 2^k>k², add inequality I proved and then proceed. You can spare me on TH-cam right?? And tell if this is right... Please? Will you marks in exam? Or in spirit of math, is the idea correct?
On my exam, i used induction twice for this problem. Once to prove since 2^n >n², if we can prove n²>2n +1 then 2^n + 2^n > n² + 2n +1 and the inequality is still true, and we get 2^(n+1) > (n+1)²
Great video! The only thing I did not understand in the demonstration is why did you replace k with 4? if the hypothesis says it must be > 4 then shouldn't k be replaced with 5? Thanks a lot.
If you plug in k=5, the inequality will not hold. We want k^2+k*k to be greater than k^2+X*k. Our original assumption is that k>4 so we have to use some X that is less than k. k^2+k*k > k^2+X*k --> X4 we can use X=4.
@@matko8038Thanks for the explanation, I was really struggling to understand why he used 4. But I have another, why do we want k^2+k*k to be greater than k^2+X*k?
@@EmperorKingKOk so, when doing induction there is a sort of rule that you aren't allowed to change one side of the equation. In this case the 2^(k+1) remains unchanged in the whole process, it is always the term on the left side. Then since we only have the right side to work with, we are trying to get the right side to create the form of some n² which is what we are trying to prove. While doing that we are free to use k>4 because that's part of our hypothesis. So to recap, we want to get (k+1)² on the right side, and 2^(k+1) must stay on the left side. Then we start working on the right side to get it into (k+1)² form, and we are starting from that first equality since we know that's true. The easiest way to get form of (k+1)² is to get form of k²+2k+1. That's why we use our hypothesis (k>4) wherever we can so we can easier get into that form, but keeping in mind if we are changing any equalities or inequalities. So finally to answer your question, It is not that we want that specifically, it's just a step that helps us get into the form we want. We already have the k² part and in this case needed the +X*k part of (k+1)²
Since (2! or 2 = 2^{1}), we can apply the laws of exponents (n^{m} * n^{n} = n^{m+n}), we can say 2^{1} * 2^{k} = 2^{1+k} & 2^{k+1}. For (n = 0) or (n = 2), we have (n^{k}) + (n^{k}) = n^{k+1} when k is a positive integer. Therefore, 2^{k+1} = (2^{k}) + (2^{k}). Although this may not be immediately obvious, it follows directly from the properties of exponents.
Math Sorcerer, U sure have done well as regards providing the pathway which clearly is referring to k > 4 very insightful of u. But here is how to put it altogether. PROOF : Let's have P(n) : 2^n > n² for n > 4 Here n = 5, 6, 7, ... Base case, when n = 5 we have, P(5) : 2^5 > 5² (true) So, P(5) is true. Assume P(k) is true for some k > 5 ==> 2^k > k² ==> 2^(k+1) > 2k² ==> 2^(k+1) > k² + k² ==> 2^(k+1) > k² + 4k (k² > 4k) Better still, ==> 2^(k+1) > k² + 2k + 2k ==> 2^(k+1) > k² + 2k + 8 (2k > 8) ==> 2^(k+1) > k² + 2k + 1 (8 > 1) So that, ==> 2^(k+1) > (k+1)² ==> P(k+1) is also true. Hence, by P M I P(n) is true for all positive integers n > 4.
@@thefunnybird7246 ... Hahahahaha, u r actually funny ... But not 2 worry, I am very much cool with very 'COMPLICATED' math though ... so 2 me, that's not a big deal at all 🙄 ... it's just part of what really makes one a Mathematician ...which is the reason why math is usually perceived 2 be tough by people who do preconceive it 2 be 'easy' ... Math is naturally complicated for it 2 remain Math ... otherwise, it really won't be Math in the first place! 👍😂
I am sorry, I must be dumber than the rest of the people here. Everything up to 5:00 makes sense. x + x is 2x. I nod along. Then 5:15 hits and you replace 2^k+2^k with k^2+k^2. ...wut? How did 2^k become k^2?? We had k^2 + 2k + 1, where did k^2 + k^2 come from??? Then 8:19 "replace this k with 4, gives you 8" ...ok... "replace the 8 with 1" WAT??? HOW DOES 8 BECOME 1???? I am sad. I will keep hunting.
What is the matter if n is a real instead of a positive integer? To demonstrate 2^n greater than n^2 for by example n >= 4.001 > 4 and not starting n=5, it has to use probably the Taylor expansion.
Thanks for the vid I've been struggling with this for ages. I'm a bit confused about where you substituted in 4 for k. How does that work like would it still cover all the values bigger than 4?
I used to struggle with your question also, tons of people do. The simple answer is that it's because k >= 4, so you can make that substitution. For example say k >= 4. And say you have 3k + p then you can write 3k + p >= 3*4 + p = 12 + p that's allowed:) You could work it out the long way. We have k >= 4, so 3k >= 12, so 3k + p >= 12 + p but nobody does that, because it's too much work. So in general, we just substitute as above.
Thank you for your help! Can you explain why k>=4 instead of k>4, because at start it define as k>4, how you change it to also be equal, or on what you depend when you say it. Thank you!
Without using “brute force”, another way of reasoning may be to compare k^2 and 2k+1. As k^2 - 2k - 1 > 0 when n > 1+sqrt(2) so k^2 > 2k+1 when n>4. This gives 2k^2 > k^+2k+1 = (k+1)^2.
Thanks a lot sir, By GOD'S Grace the problem that i have now, was being solve. Keep safe and GOD Bless Always sir. Happy Mid-Week sir. And also Praise GOD sir, Praise GOD, and also to our Lord and Saviour Jesus Christ and to the Holy Spirit who is guiding as always. And To GOD Be All The Glory Always And Forever. Amen. 🙏🙏🙏🙏. Sir.
Well ordering principle is important for those are learning proof by induction. Not sure if a video explaining this with these videos would be helpful or not. Other than that it is a good step by step proof with an excellent approach and thinking that is used in the types of proofs.
I am a bit confused. You replaced a k with 4 (I assume because that is the lowest value it can take ). Shouldn't the domain be k greater or equal to 4 in order to use four in the proof? It works with 5 as well, I am just curious as to whether this is a simple mistake or if I don't understand something. Can someone help?
First thing I thought of was proving that the equation for 2^n approaches infinity faster than n^2 using the derivative. Didnt know what induction was at the time though
Hi there, sir. I find your explanation very clear. I have a project in school, may I use this video to help students learn induction proof. Thanks for your help.
I got completely lost when you suddenly replaced 2^k + 2^k with k^2 + k^2, I have no idea how or why that was done, and everything thereafter made no sense to me. I would really love it if someone could explain what happened to me, I re-watched the video like 4 times. And since when can we just start replacing variables with numbers of our choosing? I'm so lost.
I think you made a mistake on the induction part when you replaced k with 4, because isn't it supposed to be any number greater than 4 not greater or equal to 4 ie supposed to substitute by either 5/6/7....
I see other induction inequality videos that show a different method. I find this method much more comprehensive. Would it work for all induction inequality proofs?
In the first steps you wrote n=5 because of n>4, right? And then you wrote 4 instead of k. That is to say, n>4, but not equal to 4 in this case, probably n should be > 3 and then it is ok.
So my instinct would be to pivot once you get to the “>k^2+k^2" to proving that k^2>2k+1 for all k>3. I wonder if there is any downside to that method; specifically in how that approach of basing the proof off of another lemma may fail when it is a more difficult problem and perhaps the dependency I need is harder to prove. Any thoughts?
Thank you so much. After I see the solution to a proof question that I don't know how to do, I'm always wondering to myself, "how the heck was I supposed to know to do that?" Do you have any tips?
![2^n is greater than n^2. Strategy for Proving Inequalities. [Mathematical Induction]](http://i.ytimg.com/vi/UE009vD3PEI/mqdefault.jpg)
I love this channel. Im an aspiring mathematician and frequently encounter overwhelming self-doubt about my ability. But when you explain something and reassure the audience that you struggled also, it is uplifting to know that it is not just me struggling with seemingly easy concepts.
Seriously, thank you so much for this.
I just had an assignment due today, containing this exact problem. This is a very clear way of explaining it!
Oh wow what a coincidence!!
@Kathleen McKenzie yes there mentioned that n>4, but he put k=4 , in the middle equation.....
@@ChandanKSwain yea, that confused me as well
@@TheMathSorcerer Same haha
he discovered gravity xD
You amazed me. I just came from Eddie woo and others for this question now YOU! It’s like you understand the most basic intuition needed to solve it and
you did it in so little steps. your solution is gold man. even for the factorial question. THANK YOU SO MUCH
Aww thank you!!
So true!!!!
You are good🙏🏽🙏🏽
Same, just came from Eddie
same
This is an extremely good video because you stumbled (or pretended to :) ) a couple times and talked us through how you figured it out. That’s super helpful. Thank you.
😄
Thx😄
I've been struggling a great deal in my proofs class and was self-conscious about my ability to think critically because of it. After watching this, not only do I understand the concept, I feel that I have a greater understanding of how a proof proves its claim. Thank you so much for this video, it has helped immensely!!!
Hi! 5 years later and this still rocks. Thanks internet math dude. You saved me about 2 hours of sleep. Sending good vibes.
TH-cam SHOULD OPEN A SCHOOL FOR ALL THE TH-cam TEACHERS THAT TEACH BETTER THAN SCHOOL TEACHERS. PERIOD.
@SteveEarl Watt?
@Eyosias Tewodros Are you a robot?
My ears hurt 🩸
My prof had 1hr and 30 mins to explain this topic and you nailed it within 9 mins. I understood your explanation better than my prof.
Thx, this is a hard topic to explain! I remember learning this myself and just not getting it. I ended up giving up and only understood it a year later when I looked at it again.
@@TheMathSorcerer looking again,is also a mathematical step ,it works
My teacher tried to prove instead that the difference between the inequalities is bigger than zero, I myself find that much more confusing so when I saw this, I was able to solve any problem of inequalities, thanks alot you are going to save my grades.
Thank you so much! You really inspire to continue on with school through this math stuff. Sometimes I feel very unmotivated with math because I'll try and I'll try, and when I get it, it's awesome. Plus it's something I genuinely enjoy, so it sucks sometimes when something is just not clicking. Anyhow, I've been watching some of your videos apart from the instructional math ones and they're definitely inspirational, thanks!
I appreciate the intuitive approach you take - so much of PMI instruction involves chaotic jumps in reasoning that are hard for listeners to follow and seemingly impossible to intuit ("how did you know to do that?"), so your decision to work with a problem you didn't already know is a great help. :) I was able to get this one a different way, but I had to use a pretty ugly derivative in the middle; your method is much more elegant.
For clarification, I know I am very late to responding to this video, however, when you use k=4 you must be sure that the inductive hypothesis hold for that value of k. If you plug 4 into inductive hyp it actually fails to be true. You must use a value for k that you know the inductive hyp holds true for. In this case it would need to be k=5.
yes exactly, that's the part i was confused at to why he put k= 4 when k is bigger than 4, your comment clarified me thanks
@@xreiiyoox I think he puts 4 because of < before k^2.
What are you talking about its an inequality he didnt 'plug in k=4' he replaced it! I e he threw it in the bin and replaced it with something we know for a fact is smaller than k . Since k is bigger than 4 replacing k with 4 forces an inequality it is him reshaping it so it ends up looking like the conclusion.
@@jimpim6454 yesss such a great descriptive explanation thank you!
@@TomRussle no problem 😁
7:31 "Boom" the moment of enlightenment.
This was amazing. Thank you so much. This is the 8th place I visited trying to find an intuitive explanation.
Excellent, glad I helped😃
Thank you for explaining your previous struggles with this kind of proof when you were learning. It really makes the lesson a lot more clear.
That was so cool. I am barely starting my classes for my degree and I understood nothing, but it was very cool seeing you work out the problem. Some day I’ll get it.
vashTX ... U said "some day I'll get it" ... meaning, u still haven't gotten it.
Everything's so clear now that I wanna cry oml! THANK YOU!
You're welcome!!
Fantastic! Around 5:00 you managed to easily explain what our professor has been failing to...
👍
Thank you. I was stuck on the '2^1 x 2^k' for a really, really long time. Induction is tough, and I am really overwhelmed but this video has helped me feel better.
genius! I don't know how to thank you, I was in a trouble and this video saved me, a lot of thanks again..
You are welcome😃
Love your channel. So laid back and cool. Helping me so much with my math major. Tysm!
You are so welcome!
Thank you. Hearing you talk through your thought process really helps me understand how to do this myself.
This was extremely helpful after weeks of struggling. Thank you very much. :D
Excellent!
never saw a more enthusiastic teacher on youtube 👍
Aww man, this was beautiful, you were down to earth and showed very clearly all the things I missed from too many conversations with my professor. I actually have a good idea now, of how I should think when doing these inequality proofs. Absolutely amazing. Thank you!!
2:50 /3:20 /4:47 When dinosaurs roamed the planet xDDDD
I love the humility.
These are starting to click for me and it's exciting to mess with algebra like this
thanks sir for the solution I was stuck on this question from last 3-4 hours. great help.from india.
👍
Best explanation I heard. First I thought this problem and my assignment from my pre-cal class was the same but it was actually the opposite
" Prove n^2 > 2^n for n >= 5 "
After watching the vid, I knew that the statement is already false
so how do I show that the statement above is false using The Mathematical Induction?
What a smooth proof and explanation, simply wonderful, i love induction as I loved this video!!
Think I'm a bit clear now on how to approach and tackle questions which involve proving inequalities using induction. Thanks a lot for making this video!
"when I was learning this stuff thousands of years ago..."
the stories are true. he is a sorcerer......
😂😂😂I laughed hard, oh boy🤣
I can't say how helpful this was. I will now be ready for class tommorow. THANKS!
A saving grace for discrete math this semester :)) Sets theory proofs and now I found out you do induction too, LETS GOOO!!!
I was wondering since we were working with k > 4 how you were able to substitute k = 4 into the equation. Because of the I.H it is totally plausible to do this but it would have to be k >= 4. Even for k>=4 this should work right? I assumed since k > 4 that we were only allowed to plug in 5 or greater for k since our I.H is greater than 4 not equal to it. Thank you!
I think because for the rule k>4, if we substitute k=4 in the LHS equation, then we know the LHS will be bigger than the substituted version of it because of the rule k>4. I think you can also you k=5, but then you have to use >= sign, since LHS can be bigger or equal to k=5 substituted version of it
First of all, thanks for the video everything is more clear now. Today I had my first exam at college and I had to proof that if A is a countable set then so is A^n by induction. Can you make a video of that?
Will try thank you for the idea!!
great video!! thanks for sharing your knowledge.
I have a question related to the substitution done in the minute 5:00 of the video. You said that "..you allow to do that (the substitution of 2^k by k^2) in math" and change the '=' symbol by '>'. I really want to understand how this substitution is possible and I want to know if you could provide us with any reference or material in which we could go deeper into this subject.
Thanks in advance and again, thanks for sharing.
Its because he is replacing 2^k with something he knows is smaller than it namely k^2 so obviously the equality does not hold anymore so he must write the greater than symbol.
Technically this is true for the open interval (4, infinity), so you need a more generalized induction that utilizes the well ordering relation.
How can you replace the k by 4 if it has to be >4?
he messed up there but k^2 + 2k + 10 is still > k^2 + 2k + 1.
@@DodiHD I don't think he messed up. He is not saying k=4, the inequality says > k^2+kk, so whatever is on the left side is greater than this. So using 4, we are saying it will be greater than the value obtained when substituting 4.
How you know for sure that it will be greater from the value obtained after substituting with 4?
I think it goes n>=4 because we had the exact same task like this it was only n>=5 so it's probably a mistake he didn't notice but still correct..
'CAUSE K》4.
Excellent way of explaining. Night before the submission date. Thank you Sir
You are welcome!
U are the first to teach very well me math induction thx a lot my broyher
You are most welcome!
I took discrete math 1 year ago. I didn't understand mathematical induction. This semester I am taking theoretical CS and mathematical induction is needed so I am learning it again. This is the first time I understood a proof by Mathematical inducton.
Lol me too which University
Excellent way of explaining this!! It was very helpful. Thank you! 😊
You are welcome 👍
Thank you so much for doing this video, I’ve been trying to understand this for weeks
I applied an inductive hypothesis for the original induction hypothesis and it seemed to work better
Wow, thank you so much! Excellently explained and easy to understand after you think about it a bit.
As the problem says n > 4, should we not use 5 instead of 4 in the inductive step?
At 6:42
The people want more induction proofs! Please do lots of them. (more tricky ones too)
😄👍
Thank for your explain🎉 I am studying your video from Myanmar
How I did:
Checking base case is easy...
I proved another inequality before that:
2^m>2m+1 (for m>4)
Make hypothesis and other stuff...
To proof:
2^m+2^m>2(m+1)+1 (m>4)
This reduces (by hypothesis)
2^m>2 (m>4)
Works! Nice!
Now to the main thing:
Do hypothesis and base checking...
To proof
2^n+2^n>(n+1)²=n²+2n+1
This reduces to(by hypothesis):
2^n>2n+1
Proved above!
So, hence proved. I suppose.
Is that right?
I wrote it informally...
Would do better in exam...
I should have gone the other way round like first write 2^k>k², add inequality I proved and then proceed.
You can spare me on TH-cam right??
And tell if this is right... Please?
Will you marks in exam?
Or in spirit of math, is the idea correct?
On my exam, i used induction twice for this problem. Once to prove since 2^n >n², if we can prove n²>2n +1 then 2^n + 2^n > n² + 2n +1 and the inequality is still true, and we get 2^(n+1) > (n+1)²
Great video! The only thing I did not understand in the demonstration is why did you replace k with 4? if the hypothesis says it must be > 4 then shouldn't k be replaced with 5? Thanks a lot.
If you plug in k=5, the inequality will not hold. We want k^2+k*k to be greater than k^2+X*k. Our original assumption is that k>4 so we have to use some X that is less than k. k^2+k*k > k^2+X*k --> X4 we can use X=4.
@@matko8038Thanks for the explanation, I was really struggling to understand why he used 4. But I have another, why do we want k^2+k*k to be greater than k^2+X*k?
@@EmperorKingKOk so, when doing induction there is a sort of rule that you aren't allowed to change one side of the equation. In this case the 2^(k+1) remains unchanged in the whole process, it is always the term on the left side. Then since we only have the right side to work with, we are trying to get the right side to create the form of some n² which is what we are trying to prove. While doing that we are free to use k>4 because that's part of our hypothesis.
So to recap, we want to get (k+1)² on the right side, and 2^(k+1) must stay on the left side.
Then we start working on the right side to get it into (k+1)² form, and we are starting from that first equality since we know that's true.
The easiest way to get form of (k+1)² is to get form of k²+2k+1.
That's why we use our hypothesis (k>4) wherever we can so we can easier get into that form, but keeping in mind if we are changing any equalities or inequalities.
So finally to answer your question,
It is not that we want that specifically, it's just a step that helps us get into the form we want.
We already have the k² part and in this case needed the +X*k part of (k+1)²
@@matko8038 I think I get it now (or at least I hope). You've been a great help either way. Thank you so much!
@@EmperorKingK You're welcome, please ask more of you get stuck on something :)
Sir im sorry... I still don't understand why 2•2^k= (2^k)+(2^k) at 4:40
Since (2! or 2 = 2^{1}), we can apply the laws of exponents (n^{m} * n^{n} = n^{m+n}), we can say 2^{1} * 2^{k} = 2^{1+k} & 2^{k+1}.
For (n = 0) or (n = 2), we have (n^{k}) + (n^{k}) = n^{k+1} when k is a positive integer. Therefore, 2^{k+1} = (2^{k}) + (2^{k}). Although this may not be immediately obvious, it follows directly from the properties of exponents.
Math Sorcerer, U sure have done well as regards providing the pathway which clearly is referring to k > 4 very insightful of u. But here is how to put it altogether.
PROOF :
Let's have
P(n) : 2^n > n² for n > 4
Here n = 5, 6, 7, ...
Base case, when n = 5
we have, P(5) : 2^5 > 5² (true)
So, P(5) is true.
Assume P(k) is true for some k > 5
==> 2^k > k²
==> 2^(k+1) > 2k²
==> 2^(k+1) > k² + k²
==> 2^(k+1) > k² + 4k (k² > 4k)
Better still,
==> 2^(k+1) > k² + 2k + 2k
==> 2^(k+1) > k² + 2k + 8 (2k > 8)
==> 2^(k+1) > k² + 2k + 1 (8 > 1)
So that,
==> 2^(k+1) > (k+1)²
==> P(k+1) is also true.
Hence, by P M I P(n) is true for all positive integers n > 4.
You are making it too complicated❗
@@thefunnybird7246 ... Hahahahaha, u r actually funny ... But not 2 worry, I am very much cool with very 'COMPLICATED' math though ... so 2 me, that's not a big deal at all 🙄 ... it's just part of what really makes one a Mathematician ...which is the reason why math is usually perceived 2 be tough by people who do preconceive it 2 be 'easy' ... Math is naturally complicated for it 2 remain Math ... otherwise, it really won't be Math in the first place! 👍😂
@@okohsamuel314 the crowd of the sheep looks for simple simplification, the real ones make complex things simple, and vice-versa.
Exact question came in my exam.... Thanks a lot.
Great 👍
Good. I've been in need of just this information.
Glad it was helpful!
Great video keep them coming. I remember i had the same assignament. Proof was for n>=3 in my case.
Thank you for this tutorial, I was struck with this question, and your video helped me understand. :>
Good sort of information you delivered to the viewers.
Thx
I am sorry, I must be dumber than the rest of the people here.
Everything up to 5:00 makes sense. x + x is 2x. I nod along.
Then 5:15 hits and you replace 2^k+2^k with k^2+k^2. ...wut? How did 2^k become k^2?? We had k^2 + 2k + 1, where did k^2 + k^2 come from???
Then 8:19 "replace this k with 4, gives you 8" ...ok... "replace the 8 with 1" WAT??? HOW DOES 8 BECOME 1????
I am sad. I will keep hunting.
What is the matter if n is a real instead of a positive integer? To demonstrate 2^n greater than n^2 for by example n >= 4.001 > 4 and not starting n=5, it has to use probably the Taylor expansion.
Thanks for the vid I've been struggling with this for ages. I'm a bit confused about where you substituted in 4 for k. How does that work like would it still cover all the values bigger than 4?
I used to struggle with your question also, tons of people do. The simple answer is that it's because k >= 4, so you can make that substitution.
For example say k >= 4.
And say you have
3k + p
then you can write
3k + p >= 3*4 + p = 12 + p
that's allowed:)
You could work it out the long way. We have
k >= 4, so
3k >= 12, so
3k + p >= 12 + p
but nobody does that, because it's too much work. So in general, we just substitute as above.
Thank you for your help!
Can you explain why k>=4 instead of k>4, because at start it define as k>4, how you change it to also be equal, or on what you depend when you say it.
Thank you!
@@CallBlofD I was also wondering about this. The problem states that k>4, not k>=4, so that is why I was wondering how the k could be substituted by 4
I really liked this method, thank you for your effort .
How did you replace k with 4 when you're assuming for some k>4? Aren't you supposed to replace k with a number greater than 4 because its not k >= 4?
This is amazing, I was given the first question to work out. Thanks 😍
what an explanation! I really loved it sir,
but, replacing "k" with 4 is not valid, as far as I'm concerned.
This deserves a big fat LIKE
Haha thx
Without using “brute force”, another way of reasoning may be to compare k^2 and 2k+1. As k^2 - 2k - 1 > 0 when n > 1+sqrt(2) so k^2 > 2k+1 when n>4.
This gives 2k^2 > k^+2k+1 = (k+1)^2.
Thanks a lot sir, By GOD'S Grace the problem that i have now, was being solve. Keep safe and GOD Bless Always sir. Happy Mid-Week sir. And also Praise GOD sir, Praise GOD, and also to our Lord and Saviour Jesus Christ and to the Holy Spirit who is guiding as always. And To GOD Be All The Glory Always And Forever. Amen. 🙏🙏🙏🙏. Sir.
Well ordering principle is important for those are learning proof by induction. Not sure if a video explaining this with these videos would be helpful or not. Other than that it is a good step by step proof with an excellent approach and thinking that is used in the types of proofs.
I am a bit confused. You replaced a k with 4 (I assume because that is the lowest value it can take ). Shouldn't the domain be k greater or equal to 4 in order to use four in the proof? It works with 5 as well, I am just curious as to whether this is a simple mistake or if I don't understand something. Can someone help?
I am also confused on it . You can't use 4 . We have to start with 5
I think he made a mistake, it was 5 imo.
wow, you made this problem much easier. thanks
First thing I thought of was proving that the equation for 2^n approaches infinity faster than n^2 using the derivative.
Didnt know what induction was at the time though
Thanks for sharing your method,
But there is an easier way
Please Wats the easier way
So we know that k > 4 is true in the hypothesis step. In the induction step, since n = k + 1, isn't it : n > 4 => k + 1 > 4 => k > 3 ?
Oh thanks I think k>3 makes sense
Because i was a hard time understanding why he used k=4
In the induction step and at the same time he says k>4
thank you very much. This helped me a lot :)
You're welcome!
Hi there, sir. I find your explanation very clear. I have a project in school, may I use this video to help students learn induction proof. Thanks for your help.
first time I've actually seen you do some actual math(s) but it's still big on the dry humo(u)r
thousand of years ago part is iconic
I got completely lost when you suddenly replaced 2^k + 2^k with k^2 + k^2, I have no idea how or why that was done, and everything thereafter made no sense to me. I would really love it if someone could explain what happened to me, I re-watched the video like 4 times. And since when can we just start replacing variables with numbers of our choosing? I'm so lost.
mehn.. i like the way you teach.. better than my lecturer.. lol
Thank you for your help bro. You're awesome 😎
If k>4 why do we put 4????
I don't understand why we replace K with 4 we have K is bigger than for not equal , so I don't get this point
I’m confused
real
3:20 - said every STEM major ever.
I think you made a mistake on the induction part when you replaced k with 4, because isn't it supposed to be any number greater than 4 not greater or equal to 4 ie supposed to substitute by either 5/6/7....
I see other induction inequality videos that show a different method. I find this method much more comprehensive. Would it work for all induction inequality proofs?
Yes, absolutely, the ideas are the SAME for most of these!! thank you glad it was helpful, induction inequality is so hard to learn!!
Thank you for your video. K have a question... why does the 8 becomes 1 in the last part?
Seems like maths also have exceptions!...well thankyouu so much it was understandable.
I thought K was large than 4, so shouldn't you substitute with 5 instead?
This helped me a lot
Awesome
very cool to split it up, and yeh is a great look at proofs
at 6:24 we are claiming that (k^2) + (k^2) > (k^2) + (k*k), but shouldn't those be equal??
In the first steps you wrote n=5 because of n>4, right? And then you wrote 4 instead of k. That is to say, n>4, but not equal to 4 in this case, probably n should be > 3 and then it is ok.
Thank you very much for your videos. Do you have a good book that really tackles inequalities to have a mastery in them?
So my instinct would be to pivot once you get to the “>k^2+k^2" to proving that k^2>2k+1 for all k>3. I wonder if there is any downside to that method; specifically in how that approach of basing the proof off of another lemma may fail when it is a more difficult problem and perhaps the dependency I need is harder to prove. Any thoughts?
that works but, it's also more work;) but yeah that could work!
Read my comment its easy i just proove it
Excellent tutorial indeed!
Thank you so much. After I see the solution to a proof question that I don't know how to do, I'm always wondering to myself, "how the heck was I supposed to know to do that?" Do you have any tips?
At 4:56, I don't understand why we're "allowed" to replace 2k+2k with k^2+k^2.
More please...
Will do!
So I saw the thumbnail and I was like.. wait, that's an invalid question...
So I clicked on the video and am like ooh😁. N>4
Haha yeah