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

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  • Published on Aug 9, 2011
  • Proving an expression for the sum of all positive integers up to and including n by induction
    Watch the next lesson: www.khanacademy.org/math/precalculus/seq_induction/proof_by_induction/v/alternate-proof-to-induction-for-integer-sum?YT&Desc&Precalculus
    Missed the previous lesson?
    www.khanacademy.org/math/precalculus/prob_comb/prob_combinatorics_precalc/v/birthday-probability-problem?YT&Desc&Precalculus
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Comments • 360

  • Brian Dube
    Brian Dube 5 days ago

    Amazing! Sal you're the best!

  • ItsUr Mom
    ItsUr Mom 9 days ago

    Pleass donate for khanacademy

  • SetTheCurve
    SetTheCurve 16 days ago

    He needed to spend a lot more time on that final step because I have no idea what the connection was with the original equation.

  • Cary Dominic Abejuro
    Cary Dominic Abejuro 24 days ago

    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....

  • KingUnity
    KingUnity 27 days ago

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

  • dakota gagne
    dakota gagne Month ago

    Very helpful. Made way more sense than my lectures

  • Ulitarism
    Ulitarism Month ago +2

    proof by seduction 😏

  • Іван Пилипко

    Yeahhh!)

  • OhnoItsFranc
    OhnoItsFranc 3 months ago

    Got my test tomorrow ayyy

  • Elijah Sokoni
    Elijah Sokoni 3 months ago

    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.

  • cee em
    cee em 5 months ago

    i keep seeing ppl make examples with s(k) before s(k+1) , then i see s(k-1) before s(k) , im so confused! does this work either way? can somene please give me a final answer?

  • JasonJason210
    JasonJason210 5 months ago

    It's not clear to me what the goal is with algebra after you add (k +1) at 5:42. I can see the logic of it, and how it result at 7:20 proves the proposition, but each proposition is different. I feel I need some general guidance when it comes to re-writing after the 5:42 stage. What are we after, generally?

  • maimed lord
    maimed lord 6 months ago

    long live Khan Academy!

  • Ahmed hisham
    Ahmed hisham 7 months ago

    Man everytime I'm struggling to understand something i always know that you will have a great explaintion to it thank you so much!

  • md abdool
    md abdool 7 months ago

    This video in 2019 makes me wonder how people watched these videos without 1.5x and 2x speed

  • Amber Little
    Amber Little 8 months ago

    you rock Sal

  • Rajat Chhabra
    Rajat Chhabra 8 months ago +2

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

  • [Insert Name]
    [Insert Name] 10 months ago

    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!

  • The Cold Tomato
    The Cold Tomato 10 months ago

    i remember doing this as a high school freshman and i died

  • gabi lovesinging
    gabi lovesinging 11 months ago

    Omg thk u sm ! ! ! Best explanation evrrrrrrrrrr

  • Kevin Ha
    Kevin Ha 11 months ago +1

    How about induction divisibility?

  • Anonymous
    Anonymous 11 months ago +2

    OMG YOU SOMEHOW MADE IT CLICK FOR ME YOU ABSOLUTE LEGEND

  • john holme
    john holme Year ago

    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.

  • Nsovo Ntshuxeko
    Nsovo Ntshuxeko Year ago

    Just checked In Now..18thOct2018..Limpopian

  • SHEROHE
    SHEROHE Year ago

    Butter then MIT free courses.

  • Patrik Lindfors
    Patrik Lindfors Year ago +1

    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.

  • Marius VanDamme
    Marius VanDamme Year ago

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

  • Tau Chen Yeh
    Tau Chen Yeh Year ago

    LIFESAVER!!!!!!!!!!

  • Eugene Park
    Eugene Park Year ago

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

  • Ian Mark
    Ian Mark Year ago

    i love you so much❤❤❤

  • Elena Karolína Semanová

    This genuinely makes me happy.

  • Abdullah A
    Abdullah A Year ago +1

    I understand..... nothing

  • Ernest Ng
    Ernest Ng Year ago

    man, you have no idea how many lives you saved

  • Victor Serra
    Victor Serra Year ago +2

    What i don't get about Mathematical Induction is why do you just *assume* something is true for *n* and then show it's true for *n+1* , when what you wanted to do in the first place is *prove* that thing is true for *n*

    • Joshua McNair
      Joshua McNair Year ago

      the reason is, if you can prove that it works for the base case n, which in this example is 1... and you can prove it for n+1, you can also prove it for any n, +1. You want to prove that it's true for every n, not only the first n.

    • MuffinsAPlenty
      MuffinsAPlenty Year ago

      Just restating what pcakes already said but in a different way:
      When you assume it is true for n, and prove from that assumption that the statement is true for n+1, what you're doing is proving the conditional statement: "If it is true for some natural number n, then it is also true for n+1"
      This statement, taken together with the base case, allows you to recursively work your way up to any positive integer.
      Let's say your base case is n=1.
      Since you know it's true for n=1, and since you know that if it is true for some natural number, it must be true for that natural number +1, you get that it must be true for 1+1 = 2.
      Since you know it's true for n=2, and since you know that if it is true for some natural number, it must be true for that natural number +1, you get that it must be true for 2+1 = 3.
      Since you know it's true for n=3, and since you know that if it is true for some natural number, it must be true for that natural number +1, you get that it must be true for 3+1 = 4.
      etc.
      The induction step is powerful! But, at the end of the day, it's a _conditional_ statement. It's useless without the "condition" being verified, so it doesn't work without the base case.

    • pcakes
      pcakes Year ago +3

      Victor Serra When he says "if we assume it's true for k, then we know it's true for k+1", all he means is that we know that something is true, so long as the previous one is true. He also shows that it's true for k = 1, so then we can conclude that k = 2 is true as well, since it comes right after. At that point, we can see that it's also true for 3, then 4, and so on.

  • Lixon Darvish
    Lixon Darvish Year ago

    Just made my day

  • Lixon Darvish
    Lixon Darvish Year ago

    Woooow

  • Nick Kapiskis
    Nick Kapiskis Year ago

    Thank you so much, everything is so clear now!

  • Lil Scotchy
    Lil Scotchy Year ago

    So, after confirming the base case, what you want to do is show that S(k) + (k+1) = S(k+1)?

  • Andrew Jager
    Andrew Jager Year ago

    Is it just me or are the font colours (British spelling) super juicy

  • Brian A
    Brian A Year ago +4

    why can we just assume it works for all of k?

    • SHALTILL
      SHALTILL 3 months ago

      If it works for a k then it also works for k +1 [Proven];
      It works for one [Proven];
      Then it works for two;
      Then it works for three;
      Then it works for four;
      ...

    • JUAN 12345
      JUAN 12345 5 months ago

      Thats the part I dont get. I could prove anything assuming anything.

  • poop
    poop Year ago

    Can someone please correct me if i'm wrong with my understanding:
    By proving something with induction is to first assume n=k and then by proving n=k+1 to be true.
    E.g. Killing a cow will get you beef, killing a thousand cows will still get you beef.

  • Unterseeboots
    Unterseeboots Year ago

    you're averting suicides here buddy.

  • Moneera Saleh
    Moneera Saleh Year ago

    Excuse me, are you Tom Hanks?

  • Mike Weiss
    Mike Weiss Year ago

    Awesome explanation. Really cleared this up for me. Also, I'd like you to know that when I do math, I repeat things in my head like you do during these lectures as I write them out lol, with different "pronunciations" as I repeat them. Thanks for that.

  • Awais Naveed
    Awais Naveed Year ago

    thank you so much, I was really stuck, but one thing clicked and now I understand. thank you

  • Shenella Marks
    Shenella Marks Year ago

    Honestly, n=1 and n=k are simple. n=k+1 however, is a bit confusing...

  • Mariokart360
    Mariokart360 Year ago

    I love you. Have my children

  • Good4Y0u
    Good4Y0u Year ago

    Why is there no audio?

  • RB Maths and Physics

    A bit hard to read but clear expanation

  • Toby Hang
    Toby Hang Year ago

    The check mark on point.

  • Drex Beckman
    Drex Beckman Year ago

    It finally clicked. Thank you, Sal.

  • Cha Eun-Woo
    Cha Eun-Woo Year ago

    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?

  • Gelila N
    Gelila N Year ago

    you are a life saver

  • Zeveria
    Zeveria Year ago +9

    "Assume true for K", this part pisses me off. No. You don't get to assume. I may as well say "assume 2+2= 1" and just leave it at that. This truly sounds like a bunch of nonsense.

    • Eoin Higgins
      Eoin Higgins 10 months ago

      @MuffinsAPlenty Thank you very much. Cleared up my problems with prooving by induction 🙌

    • MuffinsAPlenty
      MuffinsAPlenty Year ago +18

      Yes, you make an assumption, and that assumption may _not_ be true! This is why the base case is necessary!
      When you assume it is true for k, and prove from that assumption that the statement is true for k+1, what you're doing is proving the conditional statement: "If it is true for some natural number k, then it is also true for k+1"
      This statement, taken together with the base case, allows you to recursively work your way up to any positive integer.
      Let's say your base case is k=1.
      Since you know it's true for k=1, and since you know that if it is true for some natural number, it must be true for that natural number +1, you get that it must be true for 1+1 = 2.
      Now, since you know it's true for k=2, and since you know that if it is true for some natural number, it must be true for that natural number +1, you get that it must be true for 2+1 = 3.
      And since you know it's true for k=3, and since you know that if it is true for some natural number, it must be true for that natural number +1, you get that it must be true for 3+1 = 4.
      etc.
      The induction step is powerful! But, at the end of the day, it is a conditional statement. It's useless without the "condition" being verified. The base case verifies that "conditional" part of it.

    • Babi Salazar
      Babi Salazar Year ago

      Facts

    • Paul 123098
      Paul 123098 Year ago +2

      so a+a=a-1 ? See that doesn't work for any number except -1 because you get a= -1 for all numbers lol

  • Khokon Chowdhury
    Khokon Chowdhury 2 years ago

    it was so much helpful. thanks a lot!!!

  • Seatla Mongatane
    Seatla Mongatane 2 years ago

    Mind blown. Now to ace this section in my test

  • Kylee Webb
    Kylee Webb 2 years ago

    Can there just be a choice to take online classes through Khan instead of going to Discrete Math class at college? I would take this professor over any other professor in a heart beat.

  • KingSceptile
    KingSceptile 2 years ago

    Minecraft

  • MuffinMan
    MuffinMan 2 years ago

    Dude I love your videos; but you repeat yourself so much, so often, and so fast in this one, I'm having to fight the compulsion to rage into a conniption fit. I can't tell if it is to help us remember concepts, or because of a compulsive habit, but I'm fucking twitching in my seat. Again, I fucking love your vids. You are a great teacher. You are there for me even when I have a professor that is completely unintelligible.

    • MuffinMan
      MuffinMan 2 years ago

      I upvoted it btw. Still the best introductory induction proof vid that is reasonably searchable. I don't want you to think I hate you

  • Kyle Patrick
    Kyle Patrick 2 years ago

    Lol I know this is cliche but this is a million times better than lectures