Discrete Math - 5.1.1 Proof Using Mathematical Induction - Summation Formulae

An introduction to the method of proof using mathematical induction. Each proof in this video verifies a summation formula.

Video Chapters:
Introduction 0:00
Understanding Proof by Mathematical Induction 0:07
Prove the Summation Formula for i 2:38
Another Proof by Mathematical Induction 10:00
Conjecture a Summation Formula 15:40
Prove the Conjectured Formula is True 18:48
Up Next 23:15

Textbook: Rosen, Discrete Mathematics and Its Applications, 7e

[kletap]

I have been watching your videos all semester, you teach so well and clearly as opposed to my actual discrete math teacher. Thank you thank you so much for these videos <3

[heralds]

You have no idea how much this video has helped me. I was stressing out about doing Mathematical Induction cuz my professor just flew thru the examples in class. Especially with the fact of trying to figure what I must prove and our proof paper is on mathematical induction. But with this video and the rest of the videos, it makes so much sense! I can finally go and do the proof paper with confidence!

Thank you so much! I wish there were more teachers like you!!!

[brandon_b]

As always, your videos are a godsend. I have felt so lost in discrete math, but your videos ALWAYS break it down in an 'easy-to-digest' way. I always come away from your videos understanding better

[robdunsmuir]

I am so thankful I found you. I've been struggling terribly with my university math course. Your videos are so well done, the explanations are clear and the video goes at a perfect pace. Thank you for giving me the confidence I need to continue. I'm taking it distance ed and it's impossible without actual explaining and examples

[pandaonsteroids]

My professor does show that step! He is actually a great professor. I am a CS major so we take Mathematical Structures or Discrete Math for CS. Your videos are great for review before a quiz or test! Better than looking at my notes.

[RafidShuvo]

You taught a great way thank you ❤and your voice is also sweet

[waleedmnaeem]

disliker = the instructor who don't show the moooost important step in his video 😂😂😂

[jeremiahbarro]

At 14:10 you say that we have to add to both sides to be mathematically correct, but we didn't do that in the proof just prior. Also, this proof works without adding to both sides because
We want to show that 2^(k+1) - 1 + 2^(k+1) = 2^(k+2) - 1

2^(k+1) - 1 + 2^(k+1)
= 2 * 2^(k+1) -1 by simple algebra
= 2^1 * 2^(k+1) -1 by exponent rules
= 2^(k+1+1) -1
= 2^(k+2) - 1 which is equivalent to the right side.

This raises the question, when DO we need to add to both sides to be mathematically correct? Is it ever necessary or is it just another tool to be used for any equation?

[ToxicOsOk]

At 21:19, shouldn't it be show that 1+3+5+...+(2k-1) + (2(k+1)-1) = (k+1)^2?

Wait, nevermind... 2(k+1)-1 simplifies to 2k+2-1 = 2k+1 which is what you have.

[SuzyZou]

Such a great video, Thank you so much for the amazing explanation!

[HolyShadowNow]

I wanted to ask why you added +2 (instead of +1) at 21:13, but then realised its because we have a set of only positive integers, and that is why we had to skip 0?

[waifusaii]

Why did you stop at (k+1)(k+2)/2? I am not the best with algebra, but couldn't you have done more math? I just want to know when I should stop and move onto the next step.

[samkelly]

In the proof of the summation formula, how do you know the base case is 1 if there is no parameter indicating the domain of n? Are we just to assume it's positive integers for all general proofs of formulas?

[Hopetobebetter]

will you ever do videos on chapters like 5.1.4?

[mannyw_]

Would it be possible for you to label the videos with the chapter/lesson they're from? Thanks!

[dominicclark]

OMG. SO CLEAR. It doesnt have to be hard lol, just use english and explanations. WOW.

[oximas]

in the inductive hypothesis why do you use k and k+1 instead of n and n+1?

[miat]

Oh man I am still struggling to understand why in the summation SHOW part we have k + k+1 on the left and then we substitute the ks with k+1 on the right.