[Discrete Mathematics] Mathematical Induction Examples

In this video we discuss inductions with mathematical induction using divisibility, and then showing that 2^n is less than n!
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Hello, welcome to TheTrevTutor. I'm here to help you learn your college courses in an easy, efficient manner. If you like what you see, feel free to subscribe and follow me for updates. If you have any questions, leave them below. I try to answer as many questions as possible. If something isn't quite clear or needs more explanation, I can easily make additional videos to satisfy your need for knowledge and understanding.

[@BUILDUPMERCHANT]

mathematical induction is so confusing that sometimes you might not even realize you've got the answer😭

[@jadchaar713]

please do strong induction it is a lot harder :(

[@sharadpatel5472]

Do a video on strong induction

[@Submersed24]

I HAVE BEEN LOOKING FOR VIDS ON THIS FOR SO LONGGGG

[@y2ksteppa]

its been 7 years,how did it go

[@infinitesearch126]

Thank you TheTrevTutor

[@vaishalisontakke]

nicely explained

[@gordonchan4801]

can be proven by cases too:
case 1: n=3k
case 2: n=3k+1
case 3: n=3k+2

[@tafarir9119]

How is this done with a 3x3 matrix?

[@ianawilsonn]

Great video. Can anyone explain why n=(k+1) was used during the inductive hypothesis?

[@ReligiousHooker]

Because induction is proving that if a property is true for n, it’s true for the following n, so n+1. You can also do it like he does n = k and then prove it for n = k + 1. Or n - 1 -> n.

[@ryguy8o8]

Well If I could just do this for my proofs it would be awesome, but my professor needs more than this. So thanks for the shortcuts!

[@celebritiestrend7035]

Pls could you solve for the sum of an AP sn =n/2(2a +(n-1)d)

[@Vertek11235]

where'd he get 2^(k+1) to be equivalent to 2^k * 2? I'm so confused on that part. Can someone enlighten me
EDIT: figured it out myself, so they're equivalent because for example 2^4 = 16 & 2^3 *2 = 16, 16 = 16 checks out

[@youmustbenewhereguy]

it's like if 2^4 = 2^(2+2) = 2^2 * 2^2

[@snowa4444]

At 5:20 , I still don't understand why you added a 2 on the left side and a k+1 on the right side. Can someone explain? Thanks!

[@lu-fu]

it adds 2 on the left because 2^k+1 = 2^k + 2^1, and 2^1 = 2,
It adds k+1 on the right side because now n = k+1, when you apply k+1 on the left hand side, you also need to add k+1 on the right hand side, so n! = k! => (k+1)!, and (k+1)! = k!(k+1)

[@3e815]

big brain moment

[@kaziox3]

How do I prove that 2^n>n^2, for every n that belongs to N and n>=5?

[@cosmosisdown962]

1) basis: n=5. (2^5=32)>(5^2=25) correct. 2) n=k so 2^k > k^2. Then, n=k+1 giving us 2^(k+1) > (k+1)^2. We open the bracket using the quadratic identity and obtain 2^(k+1) > k^2 + 2k +1. Since 2^(k+1) = 2^k * 2, we can write *2^k* * 2 > *k^2* + 2k +1. From this you can see our assumption in bold. We know that the minimum amount of + (2k+1) is + 11, but we have got a *2 on the other side and doubling a number that is already greater gives us a greater value than adding an 11 to a lower number.

[@ranger0504]

could someone explain to me how k! (k+1) = (k+1)! ?

[@Nick-xz8ng]

I hope this response isn't too late.
k! = k(k-1)! = k(k-1)(k-2)! this pattern is true for all factorials.
k!(k+1) = (k+1)! because this is the reverse order of the process.

[@mayurkarmakar2519]

I have a question.Help me in solving the below question:
Q: Prove that 5^n-1 is divisible by 4. //(n is the power of 5)

[@shanolinnaidoo2977]

This reminds me of hacking...where you have to use every piece of available information to coerce the outcome. I'm still shaky on this, but slowly getting there...

[@jianma8858]

Wait in 4.48. 4! = 24? should be 10 ? ????

[@Trevtutor]

4 x 3 x 2 x 1 = 10?

[@jianma8858]

TheTrevTutor yeah sometimes lol I am sorry :)

[@faizajavedgani8030]

Do a video on strong induction

[@rixhie5485]

ang bobo ko d padn ma gets

[@staystay8386]

Seym(；`・ω・)