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!!!
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.
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.
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.
@@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.
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.
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
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!
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!!
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?
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
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?
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!
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
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?
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.
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.
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.
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?
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 RUclips right?? And tell if this is right... Please? Will you marks in exam? Or in spirit of math, is the idea correct?
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.
Since the question doesn't explicitly mention only integer values of 'n', wouldn't it be more approapriate to solve it for rationals? Induction wouldn't be possible but maybe something involving the right hand limit of 4?
it's supposed to be for integer values, oh hmm for noninteger values? I dunno I'd have to think about that one!!! Maybe what you say would work yes, not sure:) I think maybe subtracting it, and writing it like 2^x - x^2, then calling that f(x), and using some calculus, that might do it, maybe!!!
@@TheMathSorcerer I thought about it: both functions intersect at x=4, and the derivative of the 2^x term is always greater than that of the x^2 term for x>4. So it becomes trivial, I suppose. It probably doesn't make sense to make it more formal
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?
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.
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?
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.
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
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
RUclips SHOULD OPEN A SCHOOL FOR ALL THE RUclips TEACHERS THAT TEACH BETTER THAN SCHOOL TEACHERS. PERIOD.
@SteveEarl Watt?
@Eyosias Tewodros Are you a robot?
My ears hurt 🩸
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!!!
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 😁
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.
Everything's so clear now that I wanna cry oml! THANK YOU!
You're welcome!!
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.
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 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!
Thank you. Hearing you talk through your thought process really helps me understand how to do this myself.
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.
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!
never saw a more enthusiastic teacher on youtube 👍
This was extremely helpful after weeks of struggling. Thank you very much. :D
Excellent!
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.
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.
As the problem says n > 4, should we not use 5 instead of 4 in the inductive step?
At 6:42
What a smooth proof and explanation, simply wonderful, i love induction as I loved this video!!
Fantastic! Around 5:00 you managed to easily explain what our professor has been failing to...
👍
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.
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.
thanks sir for the solution I was stuck on this question from last 3-4 hours. great help.from india.
👍
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
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!
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!!
I can't say how helpful this was. I will now be ready for class tommorow. THANKS!
"when I was learning this stuff thousands of years ago..."
the stories are true. he is a sorcerer......
😂😂😂I laughed hard, oh boy🤣
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?
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
Excellent way of explaining. Night before the submission date. Thank you Sir
You are welcome!
Technically this is true for the open interval (4, infinity), so you need a more generalized induction that utilizes the well ordering relation.
U are the first to teach very well me math induction thx a lot my broyher
You are most welcome!
Excellent way of explaining this!! It was very helpful. Thank you! 😊
You are welcome 👍
At 4:56, I don't understand why we're "allowed" to replace 2k+2k with k^2+k^2.
Wow, thank you so much! Excellently explained and easy to understand after you think about it a bit.
Thank you so much for doing this video, I’ve been trying to understand this for weeks
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?
I thought K was large than 4, so shouldn't you substitute with 5 instead?
at 6:24 we are claiming that (k^2) + (k^2) > (k^2) + (k*k), but shouldn't those be equal??
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.
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
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
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 :)
Thank you for your video. K have a question... why does the 8 becomes 1 in the last part?
Hi may i ask what property or theorem you used when you replaced 2^k to k^2?
At 7:33 he has k^2 + 2k + 1. Shouldn't it be k^2 + 2k +1 + 7? If not how did he get rid of the 7?
Why 7 ?
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!!
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.
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.
Wait so how did the 8 turn into one
why can you replace k with 4?
Good. I've been in need of just this information.
Glad it was helpful!
how do you use the same method for 4^n > n^3 for all N . I try to open it up like that and got stuck at 4^(k+1)>= k^3 +3K^2 .k
Here something i don't understand , here a condition that n>4 . How to you put 2×4 replacing by 4k?
Thank you man .
Could you please tell me the size of your whiteboard ?
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
Thank you for this tutorial, I was struck with this question, and your video helped me understand. :>
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!!
This deserves a big fat LIKE
Haha thx
If k>4 why do we put 4????
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 RUclips right??
And tell if this is right... Please?
Will you marks in exam?
Or in spirit of math, is the idea correct?
Great video keep them coming. I remember i had the same assignament. Proof was for n>=3 in my case.
Good sort of information you delivered to the viewers.
Thx
Where did the 2^1 gone to?
I applied an inductive hypothesis for the original induction hypothesis and it seemed to work better
I really liked this method, thank you for your effort .
Exact question came in my exam.... Thanks a lot.
Great 👍
This is amazing, I was given the first question to work out. Thanks 😍
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.
How do we replace the 8 with 1? Why is that legal?
The lower bound is like -0.7666647 ish. What is that?
Since the question doesn't explicitly mention only integer values of 'n', wouldn't it be more approapriate to solve it for rationals? Induction wouldn't be possible but maybe something involving the right hand limit of 4?
it's supposed to be for integer values, oh hmm for noninteger values? I dunno I'd have to think about that one!!! Maybe what you say would work yes, not sure:)
I think maybe subtracting it, and writing it like
2^x - x^2, then calling that f(x), and using some calculus, that might do it, maybe!!!
@@TheMathSorcerer I thought about it: both functions intersect at x=4, and the derivative of the 2^x term is always greater than that of the x^2 term for x>4. So it becomes trivial, I suppose. It probably doesn't make sense to make it more formal
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 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
I don't know, how you placed 4 at the value of k, as it is mentioned that n >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
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.
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?
wow, you made this problem much easier. thanks
mehn.. i like the way you teach.. better than my lecturer.. lol
how can you replace 2^K with K^2?
thank you very much. This helped me a lot :)
You're welcome!
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.
How did he go from +8 to +1 at the end? I still don't follow? we're suppose to set it to equal to each other?
Okay, I think it makes slightly more sense since 8 is greater than one
Thank you for your help bro. You're awesome 😎
3:20 - said every STEM major ever.
why do we replace the 8 with 1 near the end??
I'm also confused lol
i did not get why i can replace k with 4, can someone explain to me?
thousand of years ago part is iconic
I don't get how we are writing 4 if k>4, why not 5 like in the basic step 😩 someone please explain
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
i dont really get it why reaplaced k = 4