Mathematical Induction Proof for the Sum of Squares
HTML-код
- Опубликовано: 12 янв 2025
- In this video I prove that the formula for the sum of squares for all positive integers n using the principle of mathematical induction. The formula is,
1^2 + 2^2 + ... + n^2 = n(n + 1)(2n + 1)/6
I hope this video helps someone.
![Sum of first n squares. [Mathematical Induction]](http://i.ytimg.com/vi/JG2xCiu_HDI/mqdefault.jpg)
![Felix "Unfair" | [Stray Kids : SKZ-PLAYER]](http://i.ytimg.com/vi/Oswujxm2Ag0/mqdefault.jpg)
You explained this all so well from the bottom of my heart thank you
I love the way you simplify things down on the earth. Hope you'll have a wonderful christmas and continue your math videos.
Top tier content. This exact proof is going to be in my final, thanks for the clean solution.
I like how induction makes things easier to prove but i feel a rigorous derivation best suits this theorem
Your videos are really helpful for undergraduate math courses :) Tnx from Canada
Can u make a video on integral.. To find the are under a graph why we use it's antiderivative. What does area under a function have to do with its antiderivative..
dude you're the man. thanks so much. subscribed.
Thanks for the sub!
It's a great explaination about mathematical induction thanks you🙏🏽🙏🏽🙏🏽🙏🏽🙏🏽to do for use🙏🏽🙏🏽🙏🏽
Mathematical induction is such a clever strategy.
Absolutely that helps ... thank you sir .
You are very welcome!
It's been two semesters since I used mathematical induction and I am starting to review early for my spring 2025 elementary number theory class and one of the chapter 1 sections is over PMI. I remembered in proofs I would screw up the induction step a lot especially due to the algebra mistakes
I don’t see where the squares went in 6(k+1)^2 => 6(k+1). At 6:26
thats also what I wondered? Did you figure this out?
He factored it out
I was stuck at the hard part thanks for taking me out of that 😊 guess I will have to improve my algebraic manipulations🤔
Sir I have a question there was (k+1)^2 and what happened it just became an answer
i actually have the same question
He factorized (k+1) from both k(k+1)(2k+1) and 6(k+1)^2 so it became (k+1) (k(2k+1)+6(k+1)) this also explains why he can add 6(k+1) first even though product has higher priority
To put it simply he took (k+1) from 6(k+1)^2 so it became 6(k+1)^1 only
Sorry for the messy explanation hope this helps 😂
@@aliveandhopeful5080 thank you
Why would you take (k+1) from 6(k+1)² tho? I doesn't make sense
@@weirdweed4826 Since we are trying to proof the claim that 1^2+2^2+…+(k+1)^2 is equal to (k+1)(k+2)(2k+3)/6 we need to make the proof formula to a similar form to the claim formula, that’s why we take (k+1) out from both 6(k+1)^2 and k(k+1)(2k+1) to make sure the first equation is (k+1) which is the same as the claim formula.
Ultimately our goal is to make the formula into (k+1)(k+2)(2k+3)/6 that’s why we take (k+1) out, hope this makes more sense.
thank you math sorcerer, it definitely helped me!
I just started studying physics and exactly this is on my first homework assignment
Hi , im a fan. you should add a fun CS section to boost your views. Things like how to prove that no algorithm for finding two largest elements in an array can do this in less than n + [log2 n] − 2 comparisons.
Yeah good idea!! Thank you!!
Thanks for the video!
You are welcome!
This exact proof was on my midterm and I blew it. Should have seen this video first
Great video! But I have a question regarding the 6 in the denominator. I feel like I've hit a mental brick wall or something, I just can wrap my head around where it comes from. I feel like it should be obvious or am I wrong. Could you please let me know in a reply?
Dear Math Sorcerer,
(RIP Kobe)
I wanted to ask this question in another graduate school video but I figured it'd share it here at 12:06 am 😭
I have a question, maybe it could be a video. About integrity in mathematics (ill explain). Suppose you have two students and they both have to do 20 questions in the week in their senior algebra class. They both WANT to do well but have two approaches to doing well (i.e one wants to prove to himself the other just wants the grade the fastest). So let me begin this hypothetical, support one student attempts all the questions struggles for days doing it asking for help sometimes from professor but NEVER or RARELY uses google for the answer but most importantly NEVER uses those websites that provided ridiculous solutions to everything, granted when it comes down to assignment they don't always perform the best but does put in the work each day, hour and minute (BLOOD SWEAT AND TEARS). And the other student (Student #2) attempts the question for homework each one about for 1 minute tops doesn't get the answer goes to google find that website with all the solutions learns the solutions and moves on swiftly to all the questions without any effort by memorizing answers well enough and extremely well enough for tests and exams. Now the first student clearly showed mathematical integrity but the other student doesn't. But grades wise if it turns out that the second student performs a lot better than the first despite not having the mathematical integrity to try but instead understood the system better and used it to their advantage, so it worth it to you, is mathematical integrity worth it?
I bring this up because is it worth doing the homework intensely with integrity and grit if the mark is what matters the most. And secondly, most importantly how should a student use google for help in an ethical way beyond those websites that have solutions to all math books.
I would love to hear your thoughts sometime.
And good night!
Oh good question!! I think the learning is more of the question here, who learned more, the student with less integrity or the student with more? You could make valid arguments for both cases, let's say the student who googled answers actually saw more material, this creates more exposure. At the same time the student who didn't maybe saw less problems bit worked harder on specific problems, this creates more long term learning. I think you have to find the right balance for you, and all of this while really focusing on making sure your grades are ok. Grades really do interfere with learning in soome sense although at the same time they are a great motivator. Good comments man!!!!!
Why do you drop the square on 6(k+1)??
you're factoring out k+1.. so when you take one of it out you're left with 6(k+1)
thankyou!! but I wonder, where the square from the 6(k+1)^2 go?
Aha, I see. We prove that for K, which is any +intger, .then if it is true, we move to establish K+1, and if it is true, that means every +intger will be a true result if we put it to the equation.
Because we prove for 1, and it is true. And we prove for K+1, and it is also true. All that means is K=1----> K+1=2, so it must be valid for 2 also, and if 2 is true------> K=2-----> K+2=3 AND also should be true...
because K could be any +intger. then any K+1 will be also true!!!!!
Sometimes when I look more profound at obvious things, they seem very complex and axiomatic in a fantastic way.
That was useful. Thanks
You are welcome !
@@TheMathSorcerer Thank can you do a video where you derive it though and not just prove it?
Good idea!!!
@@TheMathSorcerer yea but its fucking INPOSSIBLE to derive ive tried five different algebrsic methods and you gst TAUTOLOGIES everywhere..is that why you didnt derive it?? Tjst geometric proof doesn't count since I don't see why anyone would think of that
I found this video useful thanks.
that's the best ☺
i swear induction is soo op!
Yes!
Banger
When proving for inequality we don't do LHS and RHS and when proving for summation we do LHS and RHS right??
Good video but...I'm a bit unsatisfied with the induction proof. I would rather see a derivation of the formula. That had to come first and I wonder how it was done.
derivation left as reader exercise
@obinator9065 Fair enough. Not sure where to start however.
what happened to 6(k+1)^2?
I am scared so afraid to show my care, so I think diamonds are great and really understand what you mean or are doing. Thank you.
Thanks sir
Hi Aditya from Aditya lol
From where are u aditya?
Question to aditya from aditya.
@@adityatiwari2488 from maharashtra where are you from Aditya?
came through last minute for a comp assignment
thanks sir!
Welcome!
thanks
👊
soo confusing
smurfing
Hahahahaa
thanks sir!