Perfect Number Proof - Numberphile
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9 жыл бұрынThis video follows on from: • Perfect Numbers and Me...
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Objectivity: / objectivityvideos
Mersenne Primes and Perfect Numbers, featuring Matt Parker.
Matt is the author of Things to Make and Do in the Fourth Dimension. On Amazon US: bit.ly/Matt_4D_US Amazon UK: bit.ly/Matt_4D_UK Signed: bit.ly/Matt_Signed
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Videos by Brady Haran
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@TrackpadProductions 5 жыл бұрын
Is putting the lid on a pen the maths equivalent of dropping a mic, then?
@rogervanbommel1086 3 жыл бұрын
Lol, first at 154 likes
@2028end 3 жыл бұрын
@@rogervanbommel1086 I'm going for 155. : P
@Xayuap 2 жыл бұрын
or throwing the last piece of chalk to the rubbish trash can
@crondawg101 2 жыл бұрын
that’s called re-capping
@bryandraughn9830 Жыл бұрын
It IS! (Caps pen)
@MrDannyg77 8 жыл бұрын
Matt is great. I love his sense of humor. He's one of very few people who can take the subject of this video and make in entertaining to non-math nerds.
@andrewxc1335 8 жыл бұрын
I differentiate between groups of operations not with inflection, but with pauses: "Two to the... n minus one," versus "two to the n... minus one."
@andrewxc1335 8 жыл бұрын
It's not dramatic, or anything, just a quick stop / catch-breath type pause. Half a beat, so everyone knows what I'm doing.
@toasticide816 8 жыл бұрын
i havent had much of this experience but i generally say "one less than 2 to the n" and "2 to one less than n" rather than having "minus one" in suitable place. others find it annoying for obvious reasons but i like it. :)
@precumming 6 жыл бұрын
2 to the n ·*camera zooms in* minus one
@helloiamenergyman 5 жыл бұрын
me 2
@helloiamenergyman 5 жыл бұрын
to the n minus 1
@isaacechols2483 7 жыл бұрын
Matt becomes the child he talks about at 3:43, at 13:37
@nuralimedeuatnu 4 жыл бұрын
A nice leet reference :)
@AdamBomb5794 7 жыл бұрын
Classic Mathematician: "Let us assume that we know the total"
@numberphile 9 жыл бұрын
Matt is the author of Things to Make and Do in the Fourth Dimension. You can support him by checking out his book... On Amazon US: bit.ly/Matt_4D_US Amazon UK: bit.ly/Matt_4D_UK Signed: bit.ly/Matt_Signed
@alexroberts8755 9 жыл бұрын
I got it for Christmas, it's brilliant!
@KasabianFan44 9 жыл бұрын
Same, it was one of the best Christmas presents I ever got!
@AndresRodriguezGuapacha 9 жыл бұрын
You make me want to go back to university! Why can't all teachers be like Matt?
@guanche011 9 жыл бұрын
Hey Brady, the videos in my subscription feed listed this one before (thus older) the previous one, which made it really hard to watch.. Just something to look out for. Really great videos nonetheless!
@adithijagannadhan7174 9 жыл бұрын
It's a really good read!
@BenTheBikerBoy 9 жыл бұрын
13:39 Literally the smuggest face ever :')
@hankroest6836 5 жыл бұрын
And admittedly: "...so pleased I'm going to put the caps back on both pens." ! ;-)
@LilAnnThrax 8 жыл бұрын
It's 2am. I've become addicted to watching Numberphile before bed. I'm watching towards the beginning where we are looking at the pattern of the 2, 4, 16, 64... And I think to myself, those are powers of 2. Then I see they are the prime -1. I figure Matt will say "this is obviously just 2 to the power of the prime minus one." When he says he tortures kids with it and it's not obvious at all I feel so happy that I finally understood a non obvious Numberphile concept. I finally feel like I belong. Loved this video!
@Reydriel 8 жыл бұрын
+Ann Beckman He tortures KIDS with it, not adults, whom I believe will see the pattern pretty much immediately :P
@Cloiss_ 8 жыл бұрын
+Reydriel I'm 12 years old and I saw the pattern immediately... I'm also taking Geometry so I'm familiar with formal proofs already too.
@mikikiki 8 жыл бұрын
+EpikCloiss37 12 year olds were doing geometry in the late 1800s, too. ☺️
@Cloiss_ 8 жыл бұрын
Then what happened to our education system? Now you have to be in super special programs for that... (which are based on IQ of all things... Not a true measure in my opinion...)
@TheRedstoneTaco 7 жыл бұрын
I NOTICED THAT TOO WOW! :D I USED A CALCULATOR TO DETERMINE THAT 8191 is multiplied by 4096 to get 33,550,336!
@jz5738 9 жыл бұрын
Oh that was beautiful; math truly is the music of logic!
@tggt00 9 жыл бұрын
Usually I hear people say the opposite, music is the math of art.
@jz5738 9 жыл бұрын
I agree!
@maxischmidt1299 9 жыл бұрын
Very well said... cool^^
@oz_jones 8 жыл бұрын
+tggt00 Music is a massles body with a mathematical heart :)
@noahjames9457 6 жыл бұрын
Jasko Z Math is the science of the art of the music of logic.
@zacktobar13 9 жыл бұрын
It blows my mind how similar of a feeling this video gives me to watching my calc 2 professor do proofs for certain series tests...
@Formulka 9 жыл бұрын
"I use this to torment young people" :)
@quinn7894 3 жыл бұрын
3:27 Did he just call high school students "young"?
@wittlewill6839 3 жыл бұрын
nice
@ru2225 3 жыл бұрын
@@quinn7894 secondary (high) schoolers start at around age 10/11 in Australia and UK (where he's from and where he lives respectively), which is young :)
@golux-57 5 жыл бұрын
Matt Parker, I would have loved to have you as a math professor in school. I've always loved math, and even majored in it as college. It was teachers like you who made it even more interesting.
@dragan176 9 жыл бұрын
You should do more of these proof videos, this was really great!
@GLRaema 9 жыл бұрын
Matt looks so happy at the end of this video :D
@olanmills64 2 жыл бұрын
For some reason, I find Brady's incredulity at the beginning to be hilarious."You've already shown a link!"
@nathanielmcclaflin1374 7 жыл бұрын
I love this channel! Matt has told me everything I needed to know about perfect numbers and mersenne primes in this video and his one that came right before it that I can teach it to my classmates that know nothing about it.
@traktortarik8224 6 жыл бұрын
I just pronounce superscripts more quickly when they're together, like parentheses
@jacobbaer785 7 жыл бұрын
Haha.. "that's why Australians are so good at math" 4:54
@nathanielmcclaflin1374 7 жыл бұрын
Matt is so funny
@TruthNerds 5 жыл бұрын
"Ethan, count to ten!" "Yes, ma'am. One alligator, two alligator…" (Yes, I know there are no alligators in the wild in Australia.)
@marouaneh175 9 жыл бұрын
I would have loved to see a proof of the other way around, that is every even perfect number has a Mersenne prime factor.
@dembro27 9 ай бұрын
I think I would've gotten stuck at the geometric series step, but everything else was explained well and clicked for me. Cool!
@rocqua 9 жыл бұрын
So what about the proof that all (even) perfect numbers are of this form?
@fahrenheit2101 2 жыл бұрын
Yay! For once in my life I did the whole thing myself before watching the video. The only difference with my method was to prove the sum of that particular geometric series by induction, because I already knew what the answer was by inspection, so it seemed like the best proof to use, especially given that I didn't even notice it was a geometric series...
@NUGGet-3562 5 жыл бұрын
Holy frick I am blown away, this is one of the coolest things ever
@niansenx 9 жыл бұрын
Love it! I may need to watch it again! Any plans for a Numberphile book?
@francismoore3352 2 жыл бұрын
Omg this would be amazing!
@isaac10231 9 жыл бұрын
I saw you on tv! Outrageous acts of science! Haha that's awesome.
@hmv678 6 жыл бұрын
Fabulous proof. Thank you for a great video.
@SpiderwebRob 9 жыл бұрын
Last two vids were really good. Keep it up Brady.
@stevefrandsen7897 8 жыл бұрын
Happy 2016 Matt. I enjoy your videos.
@juandesalgado 5 жыл бұрын
Lovely video, thanks. This link was known to the ancient Greeks... but the converse (that all perfect numbers are of this form) had to wait until Euler. I wish you could dedicate one more video to this other side of the proof.
@Dombowerphoto 9 жыл бұрын
Rising inflection,,, good work
@kostal1991 9 жыл бұрын
I like this proof! Helped me to understand what was shown on the previous video.
@BritishBeachcomber 2 жыл бұрын
13:34 turns to camera, looking very smug, "but now we've managed to prove it"...
@Melthornal 9 жыл бұрын
I haven't done math in ages, but I'm proud to say not only did I follow along with the video, but I was a step or two ahead.
@KarlFFF 9 жыл бұрын
Can't wait for objectivity! The onscreen links didn't work though, but the description wasn't far away :)
@jopaki 8 жыл бұрын
"torment young people" LOL keep that up!
@stiveturtle530 7 жыл бұрын
I saw the pattern, I've never felt so accomplished
@Bo2gLiTcHmAsTeRtRoLl 9 жыл бұрын
I love these videos!
@WillFast140 9 жыл бұрын
A new year, a new Matt Parker video. What a great start to 2015! (Although I'm sure Matt would argue that a year is a meaningless or at least arbitrary measure of time)
@OmegaCraftable 9 жыл бұрын
Oh my goodness Brady is making a mausoleum channel.
@GothicKin 8 жыл бұрын
If you've ever worked with binary you know that the sum of all the powers of 2 up to n - 1 equals 2^n - 1
@TheRedstoneTaco 7 жыл бұрын
I havent even worked with binary I just learned that concept from a Khan Academy video showing how to count to 31 with your fingers xD. I feel like a special snowflake xD
@GothicKin 7 жыл бұрын
TheRedstoneTaco Or the binary number with only the nth digit =1 is exactly 2^n, 10000000.... -1 = n-1 ones, which is 2^ (n-1)
@htmlguy88 7 жыл бұрын
technically if you use both hands you could count up to over 1000 lol
@htmlguy88 7 жыл бұрын
and if you can do it with your thumbs they have 2 segments each ( some may say three including the connection to the wrist) and you get up past 1 million then.
@taysem321 4 жыл бұрын
Yes! I thought exactly that, the sum of powers up to n-1 is 1111111... with n-1 digits, and if you add 1, it becomes 1000... with a 1 and n-1 zeros, which is 2^n
@tigerbalmks 9 жыл бұрын
love you, matt and brady
@user26912 7 жыл бұрын
Isn't the pattern more clear in binary? Aren't we obscuring the pattern by thinking in decimal?
@Shadowmere29 7 жыл бұрын
But to prove that about binary, you must still use geometric series, so in the end you get the same result either way.
@JM-us3fr 6 жыл бұрын
Yes of course. This is far more easily understood in binary, so some of the algebra could be skipped, but the proof would still be necessary
@harry_page 4 жыл бұрын
6 -> 110 28 -> 11100 496 -> 111110000 8128 -> 1111111000000 The amount of 1s is n, the amount of 0s is n-1
@TorgieMadison 9 жыл бұрын
"I'm so pleased I'm going to put the lids back onto both of the pens" Hahahaha! You're good on camera! Well proof'd :)
@ripperbelgium 9 жыл бұрын
An interesting property of even perfect numbers that follows this theorem (although the proof is not as exiting) is that all even perfect numbers end with the digits 6 or 28. Another interesting fact as that the proof in this video was proven in one way by Euclides and by Euler in the other, two of the greatest mathematicians of all time. Euler also did some work on odd perfect numbers.
@leadnitrate2194 5 жыл бұрын
Actually, Euclid proved this theorem and Euler proved its coverse (that all perfect numbers are of this form.)
@KasabianFan44 Жыл бұрын
@@leadnitrate2194 That’s… literally what he said…
@leadnitrate2194 Жыл бұрын
@@KasabianFan44 I thought "one way by Euclid and by Euler in the other" meant that he was saying they proved the same thing two different ways, which isn't true. Now that you're pointing it out though, I can see how I was probably wrong.
@KasabianFan44 Жыл бұрын
@@leadnitrate2194 Ahhhhh I see, my bad
@williamsaraiva4562 3 жыл бұрын
Beautiful ❤️. Congratulations!!!
@LineGrinder01 9 жыл бұрын
Matt has got to be the best teacher ever.
@mkj1887 3 жыл бұрын
Stand and deliver?
@cbhowmick 9 жыл бұрын
thank you guys!!!!!!!
@KWGTech 9 жыл бұрын
Why math > science: You dont have idiots claiming satanism in the comments. (and just to be clear im not saying everyone is like this)
@eNSWE 9 жыл бұрын
***** lolwat. quantum mechanics is one of the most well empirically tested fields of physics there is. it has been thoroughly tested again and again and again during the entire 20th century. also, you'd be hard pressed to find any physicist at all who doesn't acknowledge it's validity.
@the0dued 9 жыл бұрын
***** are you taking about things like particle physics, super symmetry, super gravity, m-theory, super fluid vacuum theory, and loop quantum gravity. because they are not all subsets of quantum theory thought they use ideas from quantum mechanics they would be more accurately described as parts of theoretical physics.
@lvalentino6325 4 жыл бұрын
I wish to be at one of his classes🤓
@ilirllukaci5345 Жыл бұрын
Superb video.
@dominicpancella3012 2 жыл бұрын
The perfect numbers are the triangular numbers of the Mersenne primes, or the factors that you multiply by are half the prime plus 1
@Emerson_Bass 7 жыл бұрын
Matt has a book. It's called " Things to Make and Do in the Fourth Dimension Parker Square". Check it out
@burgers8 9 жыл бұрын
I've seen Matt Parker in countless numbers of these videos and I just realized he reminds me of The Doctor.
@notoriouswhitemoth 9 жыл бұрын
To avoid confusion it might help to be a bit more rigorous - and a bit more formal - with the syntax, differentiating the product of 2^n minus one from two to the power of the difference of n-1. It's a litte harder to follow, but if you understand it, it makes it clearer which is which.
@Tangobaldy 9 жыл бұрын
Totally above my intelligence! Looking forward to next video
@JM-us3fr 8 жыл бұрын
You proved each Mersenne prime makes a perfect number of that form. You should prove the converse too: every even perfect number has that specific form.
@Leyrann 4 жыл бұрын
Is that proven, or have we just not disproven it?
@shambosaha9727 4 жыл бұрын
@@Leyrann Euler proved it
@coc235 3 жыл бұрын
An odd number can't be written in that form, and we don't know if there are any odd perfect numbers, therefore this isnt proven
@Mmmm1ch43l 2 жыл бұрын
@@coc235 they specified "even perfect number" so yes, it was proven
@billstevens3796 4 жыл бұрын
And I'm screaming 256 without thinking it through, I guess I subconsciously realized it was powers of two.
@josnardstorm 8 жыл бұрын
..."negative one plus two to the n"...ambiguity gone
@stickmandaninacan 8 жыл бұрын
technically that could still mean (-1+2)^n, but i don't think any one normal would actually think that
@josnardstorm 8 жыл бұрын
+stickmandaninacan oh, yah. That hadn't occurred to me.
@ferko28 7 жыл бұрын
minus 1 plus the nth power of two is the only case that there's no ambiguity at all, i guess.
@Shadowmere29 7 жыл бұрын
+stickmandaninacan No. (-1+2)^n is 1^n, which is 1. The order that you put the base and exponent matter with this operation.
@ffggddss 6 жыл бұрын
Best IMHO is, "two to the n power minus one" vs "two to the n minus one power." Completely unambiguous. "to the" and "power" act like left and right parentheses there.
@LordMarcus 9 жыл бұрын
Is it just me, or does anyone else get a real self-satisfied kick out of people who insist it's not possible to solve infinite sums in the manner described starting at 11:00?
@steffahn 9 жыл бұрын
The sum in the video is not even infinite.
@Wout12345 9 жыл бұрын
Yeah, stuff can get a bit vague when you get to infinite sums. But this one's finite, so there's no real ambiguity to the result. The dots are not necessary, you could as well write the entire sum out and that way it's obvious all of the middle cancels out.
@screw0dog 9 жыл бұрын
This method only works for infinite sequences whose sum converges. (Unless you're a physicist who doesn't care about rigour).
@vernement4752 9 жыл бұрын
Wrong, infinity is a concept, not a number.
@BlueCosmology 9 жыл бұрын
Well, you shouldn't because they're the ones that are right. That is a perfectly valid method for solving a finite sum, however it is COMPLETELY invalid for an infinite sum other than the small subset that completely converge. Using that method you can get literally any value answer you want. Look up the Riemann series theorem. It is well known that if you manipulate an infinite sum in this way you can arise at any solution you want. For instance 1+2+3+4+... can be shown using this method to equal -50, 2, 17, 99992, 1/6 and absolutely any other value (or also equally be shown not to equal anything).
@LordNethesis 9 жыл бұрын
More of this on numberphile would be appreciated :) this is maths
@twilightknight123 9 жыл бұрын
In a previous video about the mandelbrot set and the numbers 63 and -7/4, Dr. Krieger stated that every Mersenne number (other than 63) would have a new prime divisor. Is there any way you could show a video of a proof of that? She also said that 63 being the 6th element in the sequence was the cause of it not having a new prime divisor. Is that because it is a perfect number? In that case, would the 28th element not have a new prime divisor as well? I've been struggle to find anything online proving her statement and I haven't been able to prove it myself either so if a video could be made (or at least if I could be given a link to an article) that would be fantastic.
@clickrick 5 жыл бұрын
3:35 "professional jerk". I'd love to see that as your profession on official documents.
@NickiRusin 9 жыл бұрын
Beautiful.
@Schlynn 6 жыл бұрын
Fun proof. Similar to a lot of the proofs I did when studying polygonal numbers.
@ronakpol1580 9 жыл бұрын
1st few souls to see this one!! XD don't know if it indicates how responsive my cellphones notifications are.. or how interesting these videos are that it makes me watch them even when i have a test the following day XD
@daalfredLP 8 жыл бұрын
Yeah! I found the Pattern for the Factors :D
@shush1329 2 жыл бұрын
I demand a Parker prime!
@agnesjeffery850 7 жыл бұрын
I don't change my tone when differentiating between 2^(n-1) and 2^n-1. I use pauses. There's 2 to the…n minus one vs 2 to the n…minus 1.
@hegebaggethun5650 4 жыл бұрын
Hi Numberphile, thanks for this lovely video. At time 10:10 I look at the workings and can't understand why the first line of the calculation (on top) is multiplied by (2^n) - 1, this is not included on the second line up from the bottom, did I misunderstand somewhere?
@TIMS3O 9 жыл бұрын
Another way to see that the geometric sum of 2:s at the end is equal 2^n-1 is to see the sum as a strip of n-1 1:s in binary which is the same 2^n-1
@BlaiseIgirubuntu 9 жыл бұрын
That was beautiful
@maxnullifidian 5 жыл бұрын
Watching people do math is like watching people dance - I can't do either, but it's fun to watch someone who does it well.
@WildStar2002 9 жыл бұрын
I knew that 6 was a perfect number from my childhood, but on a lonely day with nothing to do (and before the internet) I worked out that 28 was the next one and that 496 the third one when I noticed the pattern in the factors and stumbled onto Mersenne primes by accident as I tried to work out more perfect numbers. I was so excited! Alas, that I was not the first (by millennia) - but it was still fun to discover on my own! Awesome video and explanation of why it works out this way. Thanks!
@JM-us3fr 6 жыл бұрын
Mathematics at its best
@theblackwidower 5 жыл бұрын
That's always fun. I remember being bored one day and trying to write a proof for the the quadratic equasion, I think it was nearly a decade before I found out what proofs were. So satisfying.
@starponys0740 5 жыл бұрын
At 9:16, I start seeing two sequences multiplied by the Mersenne prime -- instead of just one.
@nov51947 9 жыл бұрын
I have been a fan of both Perfect Numbers and Mersenne Primes since high school (~50+ yrs ago!!), but I have never seen this proof! In the immortal words of Mr. Spock..."Fascinating!"
@sadieandbean 9 жыл бұрын
I'm in high school and I got the pattern before you said it. I do feel smug :)
@AndrewTyberg 5 жыл бұрын
Me too. I'm also in high school.
@vileguile4 9 жыл бұрын
What's à perfekt Numbers? Lol Swedish spelling correction when typing English :) What's a perfect number - the perfect question to answer at the start of this video!
@dcs_0 7 жыл бұрын
What I would give to have Matt Parker as my maths teacher...
@danphillips8530 4 жыл бұрын
The largest known perfect number, which is the 51st perfect number known, is (2^82589932)(2^82589933 - 1)
@benhbr 9 жыл бұрын
@9:09 a wrong factor (2^n-1) appears in the first line
@randomdude9135 4 жыл бұрын
Yeah
@magicalpencil 9 жыл бұрын
that hit me right in the maths
@badcalculon 9 жыл бұрын
As a CS grad, the first thing I saw was the pattern
@matobozo666 9 жыл бұрын
im sorry, but what's a CS grad?
@Slithy 9 жыл бұрын
Matej božič Computer sciences graduate, i guess.
@matobozo666 9 жыл бұрын
Slithereenn oh yeah.. probably, thanks!
@Slithy 9 жыл бұрын
Matej božič You're always welcome :)
@physjim 9 жыл бұрын
congrats i saw it in less than 5 sec and i'm still an undergrad, anyone with a basic understanding of powers can see it stop gloating, in fact if a student can't see the pattern he should be worried
@TakeWalker 9 жыл бұрын
I am in severe awe of this man's mathematical prowess.
@lexinaut 8 жыл бұрын
Knocks me Mersennesless! Two-per duper! Foundational number theory I would think. And by the way, who do you think WILL win the geometric series this year? The Common Ratios are favored. Thanks! Like this a lot!
@Fjollmongo 9 жыл бұрын
About loking smug. Matt's look at the end.. :)
@xxxromant Жыл бұрын
3:36 oh wow damn, im not sure if maybe i once already watched this video and forgot or already watched a video about it and forgot but i actually managed to figure out the pattern first try, kinda happy about that uwu
@fakjbf 9 жыл бұрын
I realized the pattern had something to do with powers of two, it was actually the first thing I saw. I just hadn't worked out what they all were before he showed it, so I didn't get the chance to see the connection to the first column.
@NoahtheEpicGuy 3 жыл бұрын
I immediately saw that pattern as 2^(n-1) because binary, 2^n (because of programming, binary is something I use on the daily), and because it related to the equation (2^n)-1, also related to binary. It's funny when you think about it, math and programming are so similar yet so different, or at least in my mind they are.
@kidbuu8025 8 жыл бұрын
you dont need geometric series to solve that, just add 1 to the 1+2+4+..., you can see that the 1 you add merge the 1, equal 2, then 2 merge 2 equal 4 and so on until it is 2 to the n, and finally minus 1 which you added earlier.
@SWhite-hp5xq 8 жыл бұрын
I went through another day not having to use these calculations, again.
@AkiSan0 9 жыл бұрын
That smug face at the end! :D
@dimitris5267 6 жыл бұрын
If we use the equation (2^n -1)(2^n-1) I noticed that the number of divisors(including the number itself) of a perfect number is always equal 2n. Does someone know why is that?
@darreljones8645 9 жыл бұрын
A little-known fact is the converse of the theorem proved here is also true: If an even number is perfect, it must be of the form described here (i.e, 2 ^ (n - 1) * ((2 ^ n) - 1) ). This was proved by either Euler or Fermat, I'm not sure which. The proof is also longer than this one.
@sethv5273 15 күн бұрын
I found the 2,4,16,64 incredibly quickly. I’m not a genius, I just had already read the top comment
@ND62511 Жыл бұрын
Interestingly enough, one way to tackle the 1 + 2^1 + 2^2 + … 2^(n-2) + 2^(n-1) summation is to write it in binary. What happens when you do that is you get a binary number that’s a series of 1s that’s n-1 digits long, so if you’re familiar with how binary numbers work it becomes immediately obvious what the sum is.
@dr.rahulgupta7573 3 жыл бұрын
Excellent presentation of the topics. Many many thanks. DrRahul Rohtak India
@Natalie-cx3cp 9 жыл бұрын
Matt, When I go to university I want to be in your class! What university do you teach at? I got your signed book for Christmas with shapes of constant width 2d and 3d, utilities mug, and the heart keyring! (I can't remember what it was called) they were the best presents ever!
@Danicker 7 жыл бұрын
Pattern at the start seemed obvious to me, but a interesting video none the less
@WorldOfDeepThought 9 жыл бұрын
There's a mistake at 10:00. It should be: (1+2+...+2^(n-1)) + (2^n -1) + ......... You wrote: (1+2+...+2^(n-1))*(2^n -1) + (2^n -1) + .........
@chevizz 9 жыл бұрын
9+10=21
@hshdhdbnd 9 жыл бұрын
Agreed, same mistake at 9:19
@Nicegeist 9 жыл бұрын
I think that was originally supposed to be a reminder, that the sum in that line adds up to (2^n)-1 ... but using commentary with round brackets in equations is not a smart thing to do.
@CYXXYC 9 жыл бұрын
***** or 9+4=30
@some1rational 9 жыл бұрын
yes, plz correct, i try to follow along but mistakes like these can literally throw the video out of wack
@ThatOldGuyYouKnow 8 жыл бұрын
Can you prove this problem via Induction of the series?
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