Proof by induction | Sequences, series and induction | Precalculus | Khan Academy

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Proving an expression for the sum of all positive integers up to and including n by induction
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[@patriklindfors2279]

This guy is so clear in everything he says. Most teachers would skip most of the stuff he's explaining because they feel it's obvious. Khan never assumes that anything is obvious and that is why his videos are so easy to follow.

[@royalbuchanan33]

Facts

[@rasibn]

Amazing

[@psyberpirate]

teachers who explain like this are the best ones! The world would be a much easier place if only teachers wouldn't assume that we know the "obvious" things in life. What is obvious to a professor, with a Ph.D. might not be so obvious to a college kid on his first day on the topic.

[@azazel70]

Straight facts

[@crimsonsm4768]

Completely disagree. He explains the problem very well, but the concept itself is impossible to grasp using this video

[@F0A0Y0N0U0S]

After muddling through my discrete structures textbook, it is so nice to find 9 minutes and 22 seconds of clarity.

[@mrinalinighosh9573]

yess true!! I finally understood in under ten minutes what the teacher couldn't do in a week

[@KatrinaSock]

You are so right. My DS text talks like a politician. It explains things and then you're really confused.

[@jonathansum9084]

This is how all my Math professors are.

[@empiremay]

agree.

[@andrewjager6177]

Unbelievable how confused I was vs how confused i am not now

[@danielgrobbelaar9672]

For what it's worth, you've managed to teach in 10 minutes what most teachers cannot do in an hour.

[@jamesspacer7994]

I just spent about an hour looking at proof by induction in an Elementary Linear Algebra book as well as some notes online from Stanford but both of those sources were a million miles away from this level of intuition! Khan Academy to the rescue! Thank you good sir, very clear, understandable and intuitive.

[@idknuttin]

proving something by mathematical induction isnt that difficult, its the question my professor assigns, he has us proving these ridiculously long sums that requires so much algebraic manipulation that it just makes the problem extremely difficult

[@materialknight]

So, to prove by induction that an equation is true for all inputs: 1. Check that it is true for the first input. 2. Write the equation, and incorporate (k+1) to both sides by following the pattern of the left-side part. For example: In the video, since the left-side part was a series of additions, (k+1) (the next number in the series of additions) was incorporated by ADDING it to both sides. 3. Transform the right-side of the above equation into the form of the original right-side. For example: In the video, the original right-side is a fraction whose numerator is the input * (the input + 1), & the denominator is 2; that's its form. And the proof merely consisted in adding (k+1) to both sides and transforming the right side into the same form of the original right side (a fraction whose numerator is the input [which in this case is k+1 instead of just n] * (the input + 1), and whose numerator is 2).

[@tubasmash]

THANK YOU FOR: the colors, the explanation, making me feel better, being great at what you do.

[@jcmschott1895]

That's what teaching should be about: making people feel confident about a subject, from the lowest level up!

[@user-cv8xu2yk7m]

As a sixth grader whose already learned some trigonometry and calculus from Khan Academy, I would agree.

[@weneverforgetwearelegion]

@@user-cv8xu2yk7mhahahaha cap

[@Dirk_Bergamot]

Yeah, talk about great..

[@moonmilk15]

Thanks again Khan, I can watch your video and understand it. You're making the world a better place. You would not believe how much better this is than my class.

[@thaiduy2346]

It has been 13 years since the video was posted, but the value it brings to new generations like me is legendary and immortal. Thank you!!

[@jonasalbaira896]

buddy you are god's gift

[@yamahrahda]

IKR!!! This, just kept me from failing math class.

[@abodalashkar5686]

@@zeyres4029 you're his mother?

[@spookimiiki5891]

fell off the wagon on my zoom course, muted the presentation, watched this at 1,5x speed and I was up to speed. Great video!

[@JalebJay]

I remember the story of how this formula was made. The creator of the proof was causing problems in class and his teacher told him that he had to find the sum of every number between 1 and 100. He started to notice a pattern when he added 1 and 100, 2 and 99, 3 and 98... seeing that he is getting 101, 50 times. So he showed it to his teacher saying the answer was 5050. The teacher didn't believe him and wrote out the whole problem and her results came out equaling 5050.

[@dacrecapstick5114]

So well explained, and i like the fact you say things multiple times! Helps it stick in my head. thank you very much.

[@xiexiefrank2]

omg this is so much better than my lectures, i question why i pay so much money to go to lectures where i get very lost. my math proofs prof sucks balls i don't understand shit when she explains but everything is so clear now that this guy explains it

[@armunalam]

you can also visit the official website (www.khanacademy.org/) to get the full list of subjects!!!!!!!!!

[@mumsazpatel9759]

4 years later and ur comment is still relevant

[@macho77vg]

@@mumsazpatel9759 6 years later and the comment is still relevant.

[@Chandler890]

the factoring out k+1 got me weak

[@abekor70]

think "FOIL"

[@williamshorrock-browne5999]

I think he's saying 'it's got him weak at the knees' as in, he enjoyed it.

[@NoActuallyGo-KCUF-Yourself]

Factoring is your friend. 😊

[@spaceghost8327]

*bruh your comment got me weak haha*

[@JavierRodriguezTX]

check out factoring my grouping

[@joealewis4121]

I should just pay you instead of paying for college courses. U explain everything perfectly.

[@StuhFu]

Thank you soo much Khan! Everybody thinks they know how to explain this but they rarely tie it up so that it makes complete sense.

[@KingUnity22]

That reveal at the end blew my mind. I didnt even realize that he had exactly rewritten the original formula.

[@elijahsokoni7997]

WOW!!! This is definitely something else. The examples are always easier than the task. We're having a test today and this is killing me.

[@tadalamanda9877]

After multiple fruitless attempts to understand this concept, I finally get it. Thanks 🙌

[@KickOff_Kreations]

This video just solved all my doubts. Always grateful for your videos.

[@nastarankianersi104]

This reminded me of the good old days back in high school when I was the straight-A kid in the class. Didn't think I would forget proof by induction someday, and didn't think I'd need it in software engineering (Automata theory course). Thanks a lot

[@fernandogameyt763]

Wow he is way clearer than my teacher. It's amazing.

[@eliyoung9406]

Lol

[@DominicI1]

Thank you. This explained proof by induction to me with the same example as my professor but 1000x easier to understand. No steps were skipped. Again, thank you

[@amberglaze909]

In my introduction to higher mathematics class, MTH 311, I can pay attention for like the first 30 mins. Then the next 30 mins I'm either staring at him write a proof on the board while thinking about what I want to get from the vending machine when class is over, or how I'm gonna even attempt to write anything on the next assessment, or anime. Or sleeping. The the last twenty mins we take an assessment where we have to write a proof on what we learned that day and two days ago.

[@Dirk_Bergamot]

As far as KZbin math tutoring videos go, not rewinding once, like watching this video: practically impossible.

[@alhussein_alahmed_lnu]

thank you very much! i have been struggling with this for the past 2 weeks now I think I get it!

[@amritraj6865]

Oh god ! I was totally intrigued about this topic. Another people were just teaching me how to solve problems based on it. None of them teaching me how it works. Hats off⚡⚡

[@zachv7449]

I don't know how they do it. I go into a video confused as shit, 5 minutes in it clicks after the video i know it like the back of my hand. Love it!

[@Skus0]

I wanna cry from the moment of understanding and clarity this video gave me after spending hours trying to understand induction from my discrete math textbook

[@eugenepark8058]

Dude. This video is about 7 years old but IT IS GOLD!!!! Thank you so much!!

[@afrojacker9]

12 years

[@CilantroJo]

You explained this better in 9 and a half minutes than my teacher did in 3 days. Mr. Khan, i love u bro.

[@jasmynedaniels5700]

I had an epiphany of understanding watching this, this was really helpful!

[@Tee0313]

You've made it so easy

[@strawberrytofu5174]

I am so lost in my discrete math course but I think I’m finally understanding thanks to this video, thank you so much

[@deladonics]

Thank you for this. I love you very much.

[@johns.8220]

My math course basically tries to regurgitate all of this in like half of one lesson- I really wish they did a full lesson on it. It's only slowing me down. This is a lot (a LOT) of help though.

[@Rinzlov]

That was clearest possible way to teach that. Thanks a lot!

[@celenial6009]

ngl this was still kind of confusing, but it really helped a huge bunch, even though this is just the math for it, without the actual proof structure. 🙏thx

[@vladaman1]

This is extremely useful in computer science and electrical engineering

[@lenardfrancis8687]

Definitely better than my school teacher... Now i understand it so well Thanx a lot khan :)

[@alex6455]

I just had the "oh my god it's clicking" moment that every student studying mathematics and science strives for. Thank you so much for this!

[@mickycheese27]

You have to substitute K+1 for all values of 'n' so it will be K+1[(K+1)+1]/2 = K+1(K+2)/2 Reason for doing this is because 'n' represents all positive integers. The reason why you can't substitute 'n' in place of 'k' is because 'n' represents any positive integer while 'k' represents a specific unknown integer. when one is added to 'k' then a new integer is formed 'k+1'. Therefore K+1 = n when P(K+1) is a function for n.

[@therealjordiano]

11k views and no dislikes, a testament to your greatness khan :) fantastic tutorial ... this looks extremely tricky to learn from a text book o_o

[@TimoKvapil]

watching this 10 years after you, with 1 million views. the impact that one 9 minute video has had is crazy

[@seyranshafie]

really helped thanks

[@vannotan1737]

OMG Nice explaination !

[@savanah2704]

As a sophomore in a college algebra class this is a godsend.

[@ninjapirate123]

how's life now

[@TheLodii]

You're saving the lives of everyone stuck with terrible professors! Thank you!

[@ninjapirate123]

hows life now

[@treyatl2006]

The binomial formula is just (x+y)^m = SUM k=0 to m; mCk*x^m*y^k. I'm not 100% sure about this but you must get it into the binomial coefficient form by letting x=n and y=0 using the binomial formula. Therefore, n^2 implies (n+0)^2 = 2C0*n^2*0^0 + 2C1*n^1*0^1 + 2C2*n^0*0^2. Hopefully this help some...

[@ruidechen]

Thanks! Really helped me. My prof has a heavy accent so I have a hard time trying to make sense of what he said when he went through this. Now I know perfectly how this works! :)

[@madhur7590]

Khan academy is the best Salman Khan is doing really good thing by providing free education online for everyone

[@ruba.-.3716]

I was first grade when he uploaded this video n now I’m in my first year college watching his videos 🤭

[@alialdinmohamad1000]

Love your content. amazing job my mam. keep up the good work!

[@wideawakeanimal1]

That very last step, fucked my mind up man! :(

[@g.guevarra]

Thanks a lot really helped me with pre calculus!

[@mbrochh82]

This is amazing.

[@nickkapiskis8872]

Thank you so much, everything is so clear now!

[@DashingAsianGuy]

we have an exam for tomorrow I know step 1 but the 2nd step induction is to hard for me and I've been diligently listening to my prof all the time. I wish there's a way for me to ace this subject and completely understand it. your video is informative though. it's getting a bit clearer now.

[@knightspygaming1287]

How did you get the formula of n(n+1) /2 , Cuz in my exam formula wasn't given. Is it question?

[@rajatchhabra3541]

You did not mention why we should use 'proof by induction' in the first place? In which situations can we use this?

[@murphy3807]

Statistics. It's how they get a standard number for probability or a constant number for algorithms.

[@justjesslyon]

I FINALLY understand. I should just watch your videos rather than go to my math lectures.

[@yost28]

Way better than my professor at the university. Honestly questions the value of higher education. Anyways thanks for this. Definitely the hardest concept in my discrete math course.

[@dakotagagne8840]

Very helpful. Made way more sense than my lectures

[@merlinthegreat100]

Khan academy pre-calc lectures + a pre-calc textbook = yet another grade of math that I will skip

[@cyanmatrix7269]

thank you khan academy for teaching me more than my prof's

[@Marius-vw9hp]

"true for two, true for three, true for four". Now THATS a tongue twister.

[@cary_domiii]

I'm now a 2nd Year Secondary Education Student Major in Mathematics and it is only by now that I've understood this topic well....

[@hellokat68]

please add correct English subtitle to your videos, it's helpful for non-native english speakers like me, thanks

[@mtado4933]

man, 2 hours with my math teacher, thanks for the help!

[@sallaklamhayyen9876]

Hi, Khan thank you so much and please do a discrete math course

[@pokemonnarutodbz1]

thank you very much. this has helped me quite a lot. I don't really have a good mathematics teacher, so I rely on this channel for tests.

[@RazaAshai]

Please do a video on strong induction

[@edwardm7292]

THISSSSSSSSSSS

[@RealEverythingComputers]

Thanks for the great video!

[@majdalkwaja9792]

thx , this is how it is supposed to be explained

[@AalejandroI]

Thank you, I had been looking for a vid on mathematical induction for quite a while :)

[@ElenaSemanova]

This genuinely makes me happy.

[@mickycheese27]

he took the common denominator '2'. Reason: When taking common denominator for (K+1), he has to multiply '2' to the numerator too so the equation stays balanced. If he didn't do that then the previous step would not be equal to the current step. All he did was added fractions with unequal denominators.

[@ermi1443]

Excellent video👏🏾

[@SynchronizedProgramming]

Wow thanks I see a lot of people use n and n-1 for the induction proof

[@bradleymorgan8223]

No matter how many math courses i've taken through the years, this man has always been there. I owe you a debt of immense gratitude!

[@WilliamFordRoom340]

It has been years since I have had to work on proofs. This makes it much easier to keep up my college math skills. Thanks!

[@johnholme783]

Thanks for taking the time to produce these videos. You have a talent for teaching mathematics and physics for the layman. Thanks to these videos I now have a good intuitive understanding of numerous mathematical and physics concepts. Keep up the good work.

[@ahnafsamin]

Finally I have understood the concept.Thank you Khan!

[@MinnesotaFarmboy]

I was able to follow the explanation until index 7:45, at which point I found it very confusing. Some things are written in yellow, and it is stated that, "well we have just proven it," ("it" is an unclear object in this case). I realized that the problem was that you haven't related "it" to the original premise by showing that every instance of k in the S(k) formula (written in green) had been replaced by K+1 at the end (written in yellow). It appears that you got excited when you got close to the end and forgot to relate it to S(k) or to show it graphically by drawing a line between them.

[@fikretcihan1747]

Congratulations. Notification: I would like to inform you that I plan to include this video among the videos reviewed in my article "Analysis of KZbinTM Videos and Video Comments on Mathematical Proof Methods". Kind regards.

[@ASLUHLUHC3]

OMG YOU SOMEHOW MADE IT CLICK FOR ME YOU ABSOLUTE LEGEND

[@omerbahat5408]

Thx helped me study for my test

[@AssAssinAddiction]

ive been watching so many videos trying to understand induction for my discrete maths class and this just broke it down so nicely for me. Thank you so much for this video.

[@lb6770]

kahn has once again saved my life (and by life I mean my test grade)

[@KeshavKhanna]

Its Khan.. If someone saves your life.. spell their name right :)

[@jff711]

Thank you very much!

[@johnparker1090]

Good explanation, I totally got it. Thanks

[@Bkaz3678]

God help me, I still don't get it.

[@rahilarshad2456]

kzbin.info/www/bejne/mn7RZqpqlN-nidk

[@aggelosspirou8815]

i dont know if u ever will see this but i hope this helps Prove that Σn= n(n+1)/2 for every possible integer n step 1 : know that something is true about Σ(1) . In our case it's Σ(1)=1(1+1)/2=1 step 2 :assume that Σ(k)=k(k+1)/2 is true step 3: since Σk=k(k+1)/2 based on step 2 we can now prove that Σ(k+1) = (k+1)(k+2)/2 explanation : Σ(κ+1) = 1+2+...+κ+(κ+1)= Σκ + κ+1 = (κ(κ+1)/2) +κ+1 ( based on step 2 )=κ(κ+1) + 2(κ+1)/2 = (κ+1)(κ+2)/2 . which is what we were looking for. But since S(n)=n(n+1)/2 is unquestionably true for S(1) ,we proved that it's true for S(2) aswell.And since now this is true for S(2) according to our proof it should be true for S(3) aswell. and it goes. hope this helped and ask anything if u want

[@j0shda116]

@@aggelosspirou8815 oof what the hell did you write

[@saranamuli9134]

thank you so much i have finally understood it but my question is whats the point? whats the point of proving by induction? why do we do it? why do we use it? i prefer to understand something than take it as it is so if anyone can answer me ... thank you

[@henri1_96]

@@saranamuli9134 Even if we have no idea why a statement is true, we can still prove it by induction.

[@fathimareeza1555]

Why do we have to add the (k+1) in to the left side of the equation? Also do have to add that (k+1) in to the left side of the equation whenever we want to prove induction?

[@pompouspineapple7622]

THANK YOU SO MUCH YOU SAVED ME THANK YOU

[@renillasubero2662]

Oh my I finally understand this clearly !!

[@RtsFps1]

When I distribute the K and the 2 to their respective (k+1) I get the same thing on both sides which is good. But I don’t understand how to factor out the two Ks to get it to look like (k+1)(k+2).

[@mosweutumelo88molfh]

much better than my lecturer

[@cp3foreverever]

lovely video, the pretty colours were very helpful

[@rhoadess]

I like the Alternative proof better, it is more intuitive.

[@Aevatheone]

Thank you for making this available and thank you so much for actually explaining each step from 0 to 1.