Discrete Math - 5.1.1 Proof Using Mathematical Induction - Summation Formulae
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- čas přidán 2. 08. 2024
- 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
Playlist: • Discrete Math I (Entir...
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!!!
disliker = the instructor who don't show the moooost important step in his video 😂😂😂
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
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@@detroitpistonsplayoffs nah, it's Ouyang
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
then why don't you attend your class
@@mr.yousifkhan7724 the university attend is 400 miles away and I’m taking the classes distance education. That’s how the entire courses offered.
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.
The hardest part is the algebra 😂😂😭😭😭
Excellent work Kim.
Thank you so much for this video!!!!
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very good video
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.
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?
I added 2 because on the left side of the equation, we started at 1 and each subsequent value increased by 2, so 1, 3, 5...n, n+2, n+4, etc.
@@SawFinMath totally missed that, thank you for explaining!
@@SawFinMath Thanks mam for answering, you saved me from this same confusion.
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?
saving my life
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?
Yes. I should have made the domain explicit but it’s all positive integers!
in the inductive hypothesis why do you use k and k+1 instead of n and n+1?
We are saying it is true for some random integer we are calling "k" and showing it is true for the next integer greater. It is a more general term than using n, which is already being used in our proof.
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.
why is there a 1 there at 6:42?
how did you get (k+1)^2 21:27 ?
Because this is for S(k+1). The rule says S(k+1) : (k+1)^2
@@semdinakt what rule?
will you ever do videos on chapters like 5.1.4?
Not for some time. My priority has to be for my own students, so I am working on content for them. Once that is complete I will be able to work on projects for my CZcams subscribers.
@@SawFinMath understandable, thank you so much for your hard work!
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Would it be possible for you to label the videos with the chapter/lesson they're from? Thanks!
Yup. I did that and linked the full playlist, as well.
IIIIIIIIIIIjustCUSSED OMG. SO ******* CLEAR. It doesnt have to be hard lol, just use english and explanations. WOW.
Hah thanks! Glad I could clear it up for you!
quick maths
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.
Sorry I didn't get back to you sooner. We are adding the next value in the sequence on the left side, so 1+2+3...+k and then the next value would be k+1. On the right side, we need to show that the new sequence would be equivalent to f(k+1)
Thank you so much
No problem!