Binomial Theorem Proof by Induction

Talking math is difficult. :)
Here is my proof of the Binomial Theorem using indicution and Pascal's lemma. This is preparation for an exam coming up.
Please let me know if I made any errors.

[@Jerros_]

Finally, a proof that isn't just 2 lines of math and then jumps to a conclusion, condensing all assumptions and steps in one go. Very neat to see you go through each step diligently!

[@quickyairsoft]

Thank you! The proof was well explained, however, if you had said "representeded" one more time I would have gone crazy haha.

[@Raselix]

Was never taught Pascal's rule so I stumbled hard on that step when I was doing this problem on my own. Most explanations didn't point out that step and moved right along. Thank you so much for explaining every step in detail!

[@JRay2113]

You're awesome! I finally got it. No many instructors/authors are explicit about the requirement to distributing Σf into (x + y) at the inductive step.

[@a3masi]

Best explanation on the web. Great work

[@katiefunk6959]

Godsend wizard man! Thanks for the help on my modern alg and number theory hw that's due in the morning 😂

[@shadow-ht5gk]

Very elegant proof, well done.

[@cameliad.b.4747]

Thanks!! The explanation is very clear. Awesome work!

[@cfire011]

Really helpful. Thanks for the awesome explanation!

[@lelosaiyan2550]

Thank you! That was a very clear tutorial.

[@jaroddavid5933]

11:03 - Explanation of Factoring k = 0 and k = m + 1

[@sxz452]

Elegant proof. Thank you.

[@sahilkhan_cs50]

Fantastic thank u very much for the proof of binomial theorem.

[@matyaslebeda7251]

insane, really well explained, thanks man

[@VictorMunch]

great explanation thanks a lot! One question: if we shift the summation index from k=0 to k=1 and m to m+1, wouldnt we also have to reduce the terms in the brackets to (m-1 *over* k-1)?

[@thinkanddo2352]

explained very well thank you.

[@RaeRae-dp3kz]

Why is the shifting of index still needed if the original index starts with 0? I'm sorry I don't understand that part very well.

[@ashishkunyal2919]

Poor is very well explained and it is very help full for me

[@SantiagoGonzalez-wy4vx]

I love this!!!

[@nikoka2980]

Thank you so much, i really needed the verbal explanation, textbooks just don't explain this problem well enough for me.

[@hussainfawzer]

What are the text books your referring to ?
Names please

[@nikoka2980]

@@hussainfawzer im referring to Czech textbooks, written by my professor - i dont think theyre translated into english

[@hussainfawzer]

@@nikoka2980
Do you have suggestions for rigorous proof based math books on these topics…
I’m mainly interested in topics such as
Binomial theorem
Series and sequences
Polynomials and rational functions
I want some suggestions to proof based books…

[@nikoka2980]

@@hussainfawzer im sorry, none written in english come to mind - but i will let you know if i ever find any

[@hussainfawzer]

@@nikoka2980
Okay

[@mangoatree]

Good job, thanks! :)

[@nohaatef7100]

Very useful . Thank you .

[@davidone8418]

you're incredible thanks

[@merlinthegreat100]

Pretty good proof.

[@zombieguy759]

I really love your videos, and I needed a favor. I need you to prove a bunch of things for me. I need you to prove the commutative property of addition for all real numbers, the multiplication of fractions, the addition of fractions, the commutative property of multiplication for all real numbers, and the distributive property for all real numbers including irrational numbers please. What I love about math is that it is always consistent and that properties are not made from thin air, and if you prove all these properties for me I will feel much better about that fact.
Please I have searched in so many places and never found a satisfying answer. Please out of the kindness of your heart answer my questions

[@lavenderjiang2002]

Thanks Ron it helps :)

[@jeanmahe8657]

good explanation congrats

[@christopheribarra1170]

You just heave to expand a binomial to a power (x+b)^n as a Taylor expansion to get the binomial theorem.

[@bradcrampton8920]

very good. thanks. Now if I can do it without watching...

[@lukschs1]

Al fin entiendo la prueba. Gracias

[@marcodonlic530]

thank you so much

[@merveguz3l]

Thank you so much

[@ericasantoyo4415]

Can you explain the rationale of how you added the x^(n+1) and y^(n+1) into the summation
Like why can we add them into the summation
Specifically why does k then begin at 0 and then n goes to n+1

[@matthewwilson4358]

a summation is just a sum of numbers, the x^n+1 and y^n+1 are just the first and last terms in that summation, that's why he rewrote them to look like the summation. You can "throw" them in because they are just terms that meet the criteria of the summation. By adding the first and last terms, you add the case when k=0 and the case when k=n+1 into the summation, because again its just addition. That's why the indices increment.

[@whatisnextthen9836]

thnx very helpful

[@shurkou]

nice video brah ty

[@mathsworld1619]

good & thank you

[@chaumlp]

6:58 Why is it k=1 and m+1? How to prove it is correct to transform from k=0 to k=1 and m to m+1? I still don't understand this.

[@AnjaliSharma06]

Thanks !

[@ShaolinMonkster]

Nice thank you

[@theviklink2044]

When you say "factor out" k=0 and k=m+1, isn't it rather that you are subtracting these terms from the sum? Because you are left with four terms and no multiplication signs in the next step, thus no factors.

[@rezaghasemzadeh9440]

very good

[@appledroplarry]

Come back to Red Alert 2. You are missed.

[@deusexmachina3496]

brilliant

[@raymondchen7716]

well done

[@dennisthompson4725]

I'm not sure what justifies changing the index at 10:50. If I'm showing that LHS=RHS how can I just change what RHS is?

[@alrafikri]

he doesn't change RHS at all. He only middling with LHS I think.

[@whatisnextthen9836]

what is pascal theorem you used

[@Hi-FiKR16]

normally it is n=k and n=0 and then you subsitute k+1

[@rickmonarch4552]

why don't you upload these pics?

[@12345papad]

Nice AMV

[@Anthony-db7ou]

Can someone explain the place thing around 12:30?

[@julienwitty6473]

He use Pascal rule which state that C(n,k)+C(n,k-1)=C(n+1,k). The goal of this is to combine the 2 summations together so we can go further in the proof. Notice that the summation have the same expression inside so now they are comparable.

[@vpambs1pt]

wow

[@Axisoft]

Ross Geller does math

[@sarthakhajirnis1908]

Awesome... but in the end it should be = RHS

[@RonJoniak]

Sarthak Hajirnis Ah, you are correct. Good catch.

[@evanurena8868]

Ron Joniak How did you obtain the summand inside the summation for the LHS to look different from the RHS.

[@RonJoniak]

Evan Urena Is there a time you are referring to?
-Ron

[@evanurena8868]

Ron Joniak Oh, never mind. You just muliplied both sides by (x+y) then simplified, am i correct?

[@Vibranium375]

@@evanurena8868 No he just broke down the exponents

[@MissAnaRichi]

It is "represented")))))

[@orlandomoreno6168]

You call summands factors...

[@mukongshu]

why don't you start from 0 at basic step? coz your k starts at 0

[@mukongshu]

+mukongshu because n is from 1, 1,2,3,4...

[@sharpnova2]

nice proof. no serious mistakes worth mentioning. handwriting a bit messy though. do you have a drawing tablet or are you using a mouse?
if it's a mouse, then props to you because it's better than my mouse-writing. but a drawing tablet might be awesome for you. i love mine. it's changed the way i teach.

[@chidionoh]

you lost me at 7:45 :(

[@daniloloma7367]

Im tired to this .....

[@VictorWLogan]

Amazing how many positive comments this guy's got for this non-explanation!! Noticing how horrible of an explanation this is, I wanted to glance through the notes. Based on what I read, I am sure that none of those people who claimed to have understood the train of thought presented here have done nothing except to confuse themselves...

[@Vibranium375]

Actually it is quite a good explanation. Even though I already knew the proof, this actually made it clearer to me.