The idea here is that you are making an even STRONGER condition, and saying that if that condition is true, the other one also has to be true You want to show: k+1
Great video :) Here is a funny trivia: in my country |N include 0, and if we want to exclude 0 from any group of number we add a*, so all natural numbers would be |N*, that means it is also easy to define the domain of 1/x which is |R* The example is great to explain induction but we can go further, it is easy to prove that n = 1, it is done in the video for n
You need to state what is accurate or inaccurate. Your comment s not helping me learn anything from you. I'd appreciate if you post something helpful 🙂
I solved the induction step like this: Considering that there exists a natural number K such that K ≤ 2^(K - 1), then K + 1 ≤ 2^(K - 1) + 1. Naturally, 2^(K - 1) + 1 ≤ 2^(K - 1) + K. But by the initial hypothesis we know that 2^(K - 1) + K ≤ 2^(K - 1) + 2^(K - 1) = 2^K; therefore, K + 1 ≤ 2^K
Thanks for this video, it really helped me learn how to better phrase some things when teaching mathematical induction, Your videos are super well done, and really stress the important points of each topic, I appreciate your awesome positive energy!
About your algebra video series You can mention about similar matrices,Cayley Hamilton theorem, and maybe Jordan form after you finish with eigenvalues you can record something about rotations , reflections , orthogonalization
Why induction works ? In my opinion it is based on structure of natural number each subset of natural numbers has minimal value each natural number has successor
It’s like a domino effect. You’ve shown that if it works for some natural number k, it will work for k+1. This in turn means that it will work for k+2 etc… (and so it will work for all natural numbers >= k ) but you’ve shown it works for 1. Thus it must work for 2, then 3, then every natural number after
Is math the most trustworthy science? Wel since it's based on certain axioms, as long as we can't prove that those axioms are true we can't really know if math is 100% trustworthy. But then again most of our systems that we have are at a certain level based on axiom(s).
if n ≤ 2^(n - 1) 2^((n + 1) - 1) = 2^((n - 1) + 1) = 2 x 2^(n - 1) ≥ 2 x n but 2 x n = 2n > n + 1 (because 2n - (n + 1) = n + 1 > 0) So we have established if n ≤ 2^(n - 1) 2^((n + 1) - 1) ≥ n + 1
By who? Induction is not "weaker" than other forms of proof. It's just as mathematically valid as other methods, if it gave less of a correct result it wouldn't ever be used
@@DBstudios98 All proofs give correct result hence the name. But there are types of proofs with their methods. As per experts, induction is weakest yet most interesting. And as per them, proof by brute force would require all exhaustive cases checked hence large complexity. Like if chess would ever be solved, it would require brute force algorithm. Even for quantum computers, solving chess would take lot of time.
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I love the diversity of the questions u cover, from simple high school to undergraduate to Olympiad material. Keep going, and never stop learning!
Guys remember to like the video, that’s the least we can do to thank him
If k+1
It wouldn't follow from that, correct.
But:
The fact that k+1
I have the same problem. Surely you can only make the right hand side bigger, not smaller.
The idea here is that you are making an even STRONGER condition, and saying that if that condition is true, the other one also has to be true
You want to show: k+1
Is it that k+1
@@danielschneider9358 you are making the condition WEAKER, not stronger. If you are trying to show that a
Great video :)
Here is a funny trivia: in my country |N include 0, and if we want to exclude 0 from any group of number we add a*, so all natural numbers would be |N*, that means it is also easy to define the domain of 1/x which is |R*
The example is great to explain induction but we can go further, it is easy to prove that n = 1, it is done in the video
for n
Nice observation. I'll take note
Sorry, I think your last step is wrong. You can only make the RHS bigger, not smaller.
Okay
why?
1:29 This is one of the reasons I like math
For examle when I used to go to school Pluto was consider a planet now it isnt
I agree
Thank you for your guidance and support. Your dedication inspires me to learn and grow.
Sincerely, [Thomas]"
The explanation is not logically accurate, so it is difficult for students to understand.
You need to state what is accurate or inaccurate. Your comment s not helping me learn anything from you. I'd appreciate if you post something helpful 🙂
Never disappoints.
Damn...I was sooo confused but my doubts are now cleared!!...Mathematical proofs can be a unforgiving but thanks to you all's well!!!
I solved the induction step like this:
Considering that there exists a natural number K such that K ≤ 2^(K - 1), then K + 1 ≤ 2^(K - 1) + 1. Naturally, 2^(K - 1) + 1 ≤ 2^(K - 1) + K. But by the initial hypothesis we know that 2^(K - 1) + K ≤ 2^(K - 1) + 2^(K - 1) = 2^K; therefore, K + 1 ≤ 2^K
Thanks for this video, it really helped me learn how to better phrase some things when teaching mathematical induction, Your videos are super well done, and really stress the important points of each topic, I appreciate your awesome positive energy!
Glad it helped!
Thank you for you message. I appreciate it too
Bro,solve this mathematical induction question 3^n≥n^2,for all natural number,I need this question answers.please!
This man is better than any class teacher and private tutor.
Totally agree
A world full of teachers like you would solve a lot of problems, not only mathematical problems!
I really love this man he is the best maths tutor I have ever seen
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what we have to prove is the same as "2n
About your algebra video series
You can mention about similar matrices,Cayley Hamilton theorem, and maybe Jordan form
after you finish with eigenvalues you can record something about rotations , reflections , orthogonalization
Your voice ❗️it literary compels me not only to learn but to understand idk how describe it but tysm for choosing to do this
very nice video !!!
Amazing can u set of questions these type in a another video
Why induction works ?
In my opinion it is based on structure of natural number
each subset of natural numbers has minimal value
each natural number has successor
It’s like a domino effect.
You’ve shown that if it works for some natural number k, it will work for k+1. This in turn means that it will work for k+2 etc… (and so it will work for all natural numbers >= k )
but you’ve shown it works for 1. Thus it must work for 2, then 3, then every natural number after
Yeah I think that’s the well ordering principle
Wonderful sir
Sorry, i don’t understand why k+1 is < or = a 2k?
There are arguments that are not logically accurate.
Wow genial esa pasión que le pone a los videos
Is math the most trustworthy science? Wel since it's based on certain axioms, as long as we can't prove that those axioms are true we can't really know if math is 100% trustworthy. But then again most of our systems that we have are at a certain level based on axiom(s).
You're the best!
For 6
if n ≤ 2^(n - 1)
2^((n + 1) - 1) = 2^((n - 1) + 1) = 2 x 2^(n - 1) ≥ 2 x n
but 2 x n = 2n > n + 1 (because 2n - (n + 1) = n + 1 > 0)
So we have established
if n ≤ 2^(n - 1)
2^((n + 1) - 1) ≥ n + 1
My faith in math is shaken by proofs such as: The sum of all positive integers is negative one twelfth.
At 5.15 u proved that 1 and 2 are natural but u did not say anything about 3
Do you have any advice for math youtubers who are just starting out ?
Thanks 🙏 I'm just having my exam in the next 24 hrs
I love your videos bro. Thanks.
One of the best videos explaining this topic
We can use Bernoulli’s inequality
The Best explanation there exist❤
To be honest I never liked proof by induction.
Among all different types of proofs, the proof by induction is said to be the weakest one actually.
By who? Induction is not "weaker" than other forms of proof. It's just as mathematically valid as other methods, if it gave less of a correct result it wouldn't ever be used
@@DBstudios98
All proofs give correct result hence the name.
But there are types of proofs with their methods.
As per experts, induction is weakest yet most interesting. And as per them, proof by brute force would require all exhaustive cases checked hence large complexity. Like if chess would ever be solved, it would require brute force algorithm. Even for quantum computers, solving chess would take lot of time.
At 11:30 I didn't see the reason for the conclusion that k + 1 = k + 1 and hence make the connection k + 1
How did you know 2k
@@PrimeNewtons you got a point there. What I could have said was that our induction step assumes that the statement k
This was cool 😎 thanks
I love your content bro
i have a question isn't 0 a natural number ? in addition to that p(n=0) is a true statement since 0
0 is not a natural number
0 is not a natural number
Beautiful ❤️
Great proof.
please help me solve this Show that
5^n +6^n < 9N for n greater or equals 2.