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.

[JRay]

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.

[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!

[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)?

[RaeRae-dpkz]

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.

[katiefunk]

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

[fatima.m]

First of all, you are AMAZING I’ve been looking for this proof for a really long time. second of all is it possible for you to make a video on the proof of the inclusion exclusion theorem using the sigma notations ? Thank you

[jaroddavid]

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

[MotjetjeTroy-bwho]

at the second last step, why did the upper bound of the sum change from (m) to (m+1) when you added Y^(m+1)??

[amasi]

Best explanation on the web. Great work

[zombieguy]

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

[nikoka]

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

[lazaredurand]

For the base case. Why don't you chose n=0 ?

[ericasantoyo]

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

[christopheribarra]

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

[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.

[matyaslebeda]

insane, really well explained, thanks man

[theviklink]

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.

[cameliad.b.]

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