The Easy Way to Add Up a Sequence

Рет қаралды 44,210

5 жыл бұрын

How fast can you add up all the numbers from 1 to 100?
Patreon- www.patreon.com/user?u=849925
Twitter- / jesseagaryt
Sources
Carl Friedrich Gauss: Titan of Science by Guy Waldo Dunnington
The Prince of Mathematics: Carl Friedrich Gauss by M. B. W. Tent
Footage (in order of appearance)
Snowden (2016)
The Imitation Game (2014)
Music
Three Hand Reel performed by NewShoe (this track is no longer available there. I downloaded it in like 2015):
/ newshoe

Пікірлер: 294

@gawainthedane3314 5 жыл бұрын

At the end I was scared he was going to whip out that skillshare/brilliant sponsorship

@felpshehe 5 жыл бұрын

Eeeeeeeeeee

@themexis 5 жыл бұрын

@@felpshehe casual advocacy is casual af

@Andlekin 5 жыл бұрын

I don't really mind edutainment channels being sponsored by for-profit educational companies.

@LuizFernando-ek9mh 5 жыл бұрын

That one OG math video you actually understand the logic behind the math

@whynotanyting 5 жыл бұрын

As a basement dweller, I accept your apology.

@NarutoOrganisation13 5 жыл бұрын

"And E..." [Ecstasy slides into view] This must have been a very pleasant video to make, then.

@somewony 5 жыл бұрын

Oh, that's Ecstasy? For some reason I thought it was an estrogen pill.

@r15u5k00 5 жыл бұрын

@@somewony so pure :,)

@NarutoOrganisation13 5 жыл бұрын

@@somewony Haha, I thought so too but I recognized the heart stamp on the pill. Kind of gives it away.

@TristanBomber 5 жыл бұрын

@@somewony Shit, I thought it was estrogen too, since "E" is slang for estrogen in the trans community. It is pride month, after all!

@Gillsing 5 жыл бұрын

I'm too innocent to recognise E. Also a bit slow on the uptake, as it took me a moment to figure out that those were all mathematical constants. I thought he was making a rebus.

@r15u5k00 5 жыл бұрын

sound effects guy should get an award

@Quroe_ 5 жыл бұрын

I can always count on you to have the best sponsorships. I never skip 'em.

@vanillaannihilation5871 3 жыл бұрын

Also the best sound effects.

@TheBlessingo 5 жыл бұрын

The production value is amazing! I love your work

5 жыл бұрын

Love the sound effects while doing calculations!

@haomakk 5 жыл бұрын

ok, now THIS is the best channel on KZfaq

@AusReddit 5 жыл бұрын

This method works for sequences going by two's as well (2,4,6,8...). Just divide the (last number) - (first number) by another 2 and you've figured it out. As a Mathcounts competitor, knowing this method was a MUST. Either that or you had some extremely fast calculators.

@PopeGoliath 5 жыл бұрын

This episode has been brought to you by: Drugs! Yay!

@ReasonMakes 5 жыл бұрын

Actually the best KZfaqr there is

@martinshoosterman 3 жыл бұрын

I don't know why I just watched this video. I'm a 4th year math student at university and I've known about this since the 10th grade. (Not because I figured it out, but because numberphile made a video about this like 8 years ago) But I love this result and this is one of the most well made videos about this result that ive seen.

@sorenkair 5 жыл бұрын

Didn't mention the equation works with both even and odd number of numbers.

@ericathefae 5 жыл бұрын

Yeah, I was wondering that - would have loved to hear the explanation for odd numbered lists...

@ericathefae 5 жыл бұрын

@@Errenium Thank you!

@icedragonaftermath 5 жыл бұрын

Well, the unpaired number in a sequence of an odd length would have to be halfway between the highest and the lowest values. So the unpaired number should be half * (highest + lowest). The original equation takes that into account though.

@tommyzheng387 3 жыл бұрын

Caleb Johnsen yeah and there’s a pattern

@RedrunLoL 5 жыл бұрын

I'm loving the more frequent uploads

@Gregorsnek 4 жыл бұрын

this aged poorly...

@genessab 5 жыл бұрын

I don’t know why but I find your content so insanely hilarious. I keep laughing wildly on the train

@jasonpeng5798 5 жыл бұрын

These videos are so satisfying and interesting. They are good to pass time and also I feel like I'm not totally wasting my time. Please make more.

@jazzboots8893 5 жыл бұрын

I am never ambivalent about your videos! Always great!

@Goblin4Coin 5 жыл бұрын

Every video you upload is a delight

@thror1709 5 жыл бұрын

This is great content! glad i found your channel

@Jaur-jaur 5 жыл бұрын

OMG haven’t watched the video yet but I already love it!! Yay this channel is B-A-C-K!

@JellyWaltzov 5 жыл бұрын

this made my day. perfect explanation. Thank you!

@trulyUnAssuming 5 жыл бұрын

Nicely swept the uneven numbers under the table :-p Just in case someone is interested: For uneven n, 0,...,n are n+1 numbers and n+1 is even. So you get (n+1)/2 pairs with value n, which adds up to (n+1)*n/2, which is the same formula as for the even numbers, where the n/2 pairs add up to n+1. And since you can add the numbers up to some n, the formula for adding up numbers between two numbers is just subtracting the ones from 1 which you don't want to add. I.e. the numbers between m and n are the numbers up to n minus the numbers up to m-1. So you get (n+1)*n/2 - m*(m-1)/2=((n+1)*n-m*(m-1))/2

@wepranaga 5 жыл бұрын

when you see math as a problem solving rather than counting.

@NateandNoahTryLife 5 жыл бұрын

In 2019 I’m still dumber than a 7 year old from 1784... and that’s ok.

@GoneZombie 5 жыл бұрын

We're all dumber than Gauss, tbh.

@thesuperdoge2476 4 жыл бұрын

Hey at least we're still alive

@deidara_8598 3 жыл бұрын

You have the world's knowledge at your fingertips, you can undo this whenever you like.

@misterkid 5 жыл бұрын

I'm so glad I subscribed to you

@ADPuckey 5 жыл бұрын

Jesse I'm so glad you're back. The E at the end made me lose my shit

@kadenceboatman927 5 жыл бұрын

Great, as always

@gabrielcavuquila4380 4 жыл бұрын

This is actually geometric progression but in a simpler way and I loved it, it would help a lot of my colleagues back in 10th grade

@DougStoddart 2 ай бұрын

great video - well done!

@deidara_8598 5 жыл бұрын

The key to making observations like Gauss did is to think more abstractly about the problem you're solving, rather than use pre-meditaded formulas and equations you've memorized.

@Goodvvine 5 жыл бұрын

I wish you can keep doing these videos, because they are really good

@defalt9405 5 жыл бұрын

Im so glad that you came back to KZfaq

@themasstermwahahahah 5 жыл бұрын

Keep up the good work sir!

@czechm4te 5 жыл бұрын

Thank you my brain expanded from smol brain to big brain thank

@thewarmedic2330 5 жыл бұрын

As a UIL student I appreciate it a LOT

@tverdyznaqs 5 жыл бұрын

Huh, I didn't expect to laugh at a maths video! Good job dude, this was perfect

@videogyar2 5 жыл бұрын

I realized this when I was around 13. It was a similar a question in a regional math test and I was so proud to come up with it lol

@maxgamesst1 5 жыл бұрын

And then they clapped

@macskasbogre133 5 жыл бұрын

@@maxgamesst1 It's not a farfetched claim...

@driveasandwich6734 3 жыл бұрын

To think there was a time were kids could just be the discoverers of mathematical properties by just fiddling around...

@gangstabib 5 жыл бұрын

Keep up the good work! This guy did it when he was a 7 year old kid!

@whynotanyting 5 жыл бұрын

"This video is sponsored by pie, tao, and e." hmmm pietaue? Oh, π τ e I'm going to bed

@Supernoxus 5 жыл бұрын

I love your vids

@fr4ggle4 5 жыл бұрын

New episodes of This Place?? Yes plz and thank you

@mb98765 5 жыл бұрын

don't know how or when i subscribed, but i enjoyed the video

@benjaminjackson8663 5 жыл бұрын

Same

@pabloaragon3303 5 жыл бұрын

Same

@marsherr 5 жыл бұрын

You are amazing

@WangleLine 4 жыл бұрын

These videos are so good. I hope you'll return some day~

@Swordfish42 5 жыл бұрын

Make more of this sweet, sweet content. Please.

@awsomebot1 5 жыл бұрын

I already knew about this formula and it's story, but I've still watched this video multiple times. Please don't go again :^)

@ITR 5 жыл бұрын

You can also do (last + last^2 + first - first^2)/2

@owllover2319 5 жыл бұрын

2:02 Counting is for Philistines!

@nodisponible8 5 жыл бұрын

love that endings

@imaytag 4 жыл бұрын

Jesse where did you gooooooooo......?

@platonic4ssploughing 4 жыл бұрын

ok idk if im going crazy or not but i swear there was a video from this channel that had a section that critiqued a highschool textbook for its poor wording and credibility and had this giant ass wheel of fallacies where he shortly then made a gag about him never falling for these fallacies out of sheer superhuman perfection

@nesrine7738 4 ай бұрын

Omg thank you so much, i finally get it✨✨✨😭🕊️

@Matthew-yn8pv 5 жыл бұрын

very nice

@tomoyahiro7036 5 жыл бұрын

I just came here to figure out how he explained the method even though I have a prior knowledge about the use of arithmetic series (sum of the arithmetic sequence) formula.

@murphygreen8484 5 жыл бұрын

Wait. When did I subscribe to this channel? Guess I'll unsu......whaaa? Good video. Guess I'll stay subscribed!

@scrubby2 5 жыл бұрын

i'm glad this works on odd amount of numbers too.

@TTbelis 5 жыл бұрын

I love this thnk you

@jotage3446 4 жыл бұрын

I miss this channel being active :(

@hurktang 5 жыл бұрын

Note that it's the mean value multiplied by the number of items. This is how calculus work.

@Vaaaaadim 5 жыл бұрын

I think that would be more a statistics thing

@LyubomirIko 5 жыл бұрын

ah this guy upload again

@hoomanvassef6483 3 жыл бұрын

Another way to work it out visually/intuitively is to think of the numbers forming a digital "triangle", e.g. (0) + 1 + 2 + 3 + 4 where the numbers are represented by os: xxxx oxxx ooxx ooox oooo But you note the xes also form the same "triangle", and the 2 combined form a rectangle of size 4 x 5, i.e. n [last number] x (n + 1). Hence 1 + 2 + ... + n = n(n+1)/2. For the more general case m + (m+1) + ... + n, the numbers form a digital "trapezoid", made up of one n x (n - m + 1) rectangle, with a digital "triangle" removed from it that's 1/2 of an (n - m) x (n - m + 1) rectangle. => m + (m+1) + ... + n = (n - n/2 + m/2)(n - m + 1) = (n + m)(n - m + 1)/2

@connorking8503 5 жыл бұрын

It works for odd-length sequences. All on it's own. Magic!

@Vaaaaadim 5 жыл бұрын

Indeed, it does happen to work for odd-length sequences, but even so it is worthwhile to show that this is true as well, because this doesn't always work out in general. Like for instance, there is a pattern that holds for the cosine power reduction formulas on the even powers that doesn't on the odd powers (but they are almost the same).

@lindhe 5 жыл бұрын

You get extra points for the sound effects.

@rudeus6621 3 жыл бұрын

3:14 that sound effect tho😂

@nicolasyan1613 5 жыл бұрын

Good mnemonic/derivation for the sum from 1 to n: make a list of rows with 1, 2, 3, ... n dots in each: (illustrated for n = 4) o oo ooo oooo Then, add on the same sequence in reverse to make a rectangle: o xxxx oo xxx ooo xx oooo x Since it's a rectangle, the number of dots must be n(n+1) (the number of rows is n, the number of columns is n+1). Since you added two of the sequences you wanted to count up, you have to divide by two. So 1+2+...+n = n(n+1)/2

@CoryMck 5 жыл бұрын

I figured this out in 3rd grade... _but I didn't get a story credited to my discovery_

@PeterLiuIsBeast 3 жыл бұрын

If you think about it, it's also quite like the formula for the area of a trapezoid (or triangle with top side = 0).

@splitscim 5 жыл бұрын

Did anyone else come up with the equation before he told the story? Good practice for AP Comp Sci! Seems like an FRQ they'd put on the test

@Devynwithawhy 5 жыл бұрын

I used this for a level up system in an rpg that I stoppex working on

@Cat_in_Spacetime 5 жыл бұрын

Credits - Jesse Agar

@Rovsau 3 жыл бұрын

**Mathematicians vigorously refresh their ink pens**

@Wezla 5 жыл бұрын

Love it

@raziphaz2219 5 жыл бұрын

He has risen

@sk8rdman 4 жыл бұрын

I came across this same problem, but the way I solved it was that i imagined having a bunch of blocks, one for 1, two for 2, etc up to n. If I stack those blocks on top of each other, I can make them into a right triangle. Then to find out the total number of blocks, I just solve for the area of the triangle, which is half the area of an n*n square. The only nuance you have to worry about is that there's one extra row in the middle that adds an extra n/2 blocks. So my formula was (n^2+n)/2, which is algebraically the same as your formula. So for n=100 I just take (10000+100)/2=5050

@blueicer101 5 жыл бұрын

He’s right. I figured this when solving a maths problem in year 8. How many presents do you get from your true love in the 12 days of Christmas?

@L4Vo5 5 жыл бұрын

Can you make more videos where you solve math with sound effects?

@gecko2000405 4 жыл бұрын

What sound does the equation make when the numbers work themselves out again?

@hero19876 5 жыл бұрын

Glad you aren't deceased

@xexpo 5 жыл бұрын

Would've been nice if you went slightly further with the sigma notation. For the 534->6389, you might've showed that you can do (1->6389) - (1->533), showing that the simplified n formula can be used for all.

@TheReaverOfDarkness 5 жыл бұрын

That would have been a third formula which is simpler than the first but more complex than the second.

@ArtArtisian 5 жыл бұрын

Btw, i found the history of this story of gauss really interesting. Sorta a lense for how mathematical fables have changed over time. The original sum was probably shorter, the intent of the school teacher has been totally implied, and I'm pretty sure an early version mentions the risk of violence for students. See the following for a collection of versions, dated. bit-player.org/wp-content/extras/gaussfiles/gauss-snippets.html

@leocossham 5 жыл бұрын

You learn how to do this in AS level maths in the UK

@danielsteel5251 5 жыл бұрын

This reminds me of when minutephysics did a FOIL video.

@leocossham 5 жыл бұрын

@@danielsteel5251 wait like foil as in what you wrap your sandwiches in or am I missing something

@danielsteel5251 5 жыл бұрын

@@leocossham en.wikipedia.org/wiki/FOIL_method

@leocossham 5 жыл бұрын

@@danielsteel5251 haha I understand the parallel there 😂😂 I remember this being taught as the smiley face method because you can sort of make the lines look like a smiley face?

@carlgroenvald5250 5 жыл бұрын

Gauss var FIRE år gammel

@ksmyth999 4 жыл бұрын

Do we know definitely that Gauss used the pairing method since in Wikipedia there seems to be some doubt? Also, his age is variously quoted as from 7-12. Many years ago I seem to remember reading that he was 10. Gauss was a child prodigy who could do complicated calculations in his head. This does not necessarily mean he was good at maths. But this story clearly demonstrated his promise, and he is now recognized as one of the greatest mathematicians of all time. There is another fairly simple method that he could have used. It is not quite so elegant but takes advantage of the symmetry in the numbers 1 to 100. I think he would have found this first and although it is a two-stage calculation if you include the thinking time, I believe he could have produced the correct answer just as quickly. It is fairly obvious, so I leave it as an exercise to the reader to work out.

@samar5838 4 жыл бұрын

Dude I learnt to do this in 10th grade.. its a simple arithmetic progression. You can even apply the formula n(n+1)/2 where n is the number of natural numbers taken starting from 1

@itisdevonly 5 жыл бұрын

Match number pairs. 1 and 99, 2 and 98, 3 and 97, etc. That covers numbers 1-49 and 51-99 to make 49 number pairs that each add to 100, putting the total so far at 4900, then add in the 100 and 50 that were left out, and you get 5050. Did that calculation in a few seconds. Did I get it right? Maybe I don't need to watch past the 0:18 point if I came up with that already? Edit: Okay, finished watching the video. I wasn't far off. The solution he showed was a bit more general purpose than what I did, but it was pretty much the same idea.

@AnDr3W91 5 жыл бұрын

So what was that first movie he talked about?

@blue6305 5 жыл бұрын

"No, bitch." ~Carl Gauss

@karthikkumar6861 4 жыл бұрын

Hmm, I learned it slightly differently. Sum of any 'n' no of digits in a series is always (average X no. of digits). To find the average for a series you can do (First digit + Last digit)/2. So for this question Average would be (1+100)/2 = 50.5. Now sum would be 50.5 X 100 = 5050.

@Axian_empire Жыл бұрын

Thank god. This video helped me survive my Asian parents

@WilliamBoothClibborn 5 жыл бұрын

I pray that this summation of series comes up in the further maths a level paper tomorrow.

@emanueldobos8452 5 жыл бұрын

Good luck!

@ledgeri 5 жыл бұрын

Almost skipped the sponsor ad, by routine! (Also funny: 12 is the view-count, 10 is the comment-count :) )

@Sh4d0wch40s 5 жыл бұрын

I don't get it, no matter how I think about it. Pi tau e... pie towel e? Explain... it annoys me.

@TheReaverOfDarkness 5 жыл бұрын

@@Sh4d0wch40s I don't think there's a hidden meaning in it. Instead of having an actual sponsor, he's just mimicking other youtubers while listing some fascinating numbers and trying to twist them into a pun for a visual image. He must have saved e for the end because he didn't come up with a pun for it, and also the heart just means he loves us. So basically it doesn't have one meaning, it has a lot of meanings, and they're all pretty simple. Welcome to the mind of Jesse Agar.

@therattleinthebook397 5 жыл бұрын

They backq

@gabes6108 5 жыл бұрын

But the even more generic form of the problem when the gap sizes vary is (LastNumber)=L and (FirstNumber)=F and (Gaps)=G is (L^2-F^2+L+F)/(2G)

@Vaaaaadim 5 жыл бұрын

And even more generic than that is to find the sum of a polynomial, and there are ways to come up with formulas for that as well.

@gabes6108 5 жыл бұрын

Where would one find this formula or a collection of them.

@Vaaaaadim 5 жыл бұрын

@@gabes6108 As it happens, this is a problem that I have had interest in for a very long time, and I had figured out multiple ways you can derive the formulas for this. There is a lot that I want to talk on this subject, and I intend to make a video about it one day. I'll show you what I think is the most straightforward way I've come up with. Note that I will only be talking about polynomials with integer powers. We are looking for a formula for the summation of a polynomial. Since a polynomial is a sum of powers of x multiplied by coefficients, we can split up the polynomial into its terms as separate summations and factor out the coefficients. So our problem is now essentially reduced to, can we find a formula for the summation from 1,...,n of x^p? If p = 1, we have an arithmetic sequence, p = 2 the perfect squares, so on so forth. The key observation is that, if we have some polynomial p(x), and we describe another function s(x) = the sum of the evaluations from 1,...,x of p(x), then it MUST be the case that s(x) - s(x-1) = p(x), and s(1) = p(1). Anyways, we can actually make a unique polynomial s(x) which can satisfy these conditions. Consider if I said s(x) = x^m, then s(x) - s(x-1) = x^m - (x-1)^m, and if you apply binomial expansion you will find that the largest powers cancel out and you get a polynomial of ONE LESS DEGREE, and furthermore, the highest powered term in this polynomial is m * x^(m-1). Cool, so lets use m = p + 1, and divide by p+1. But now we have all this other junk, x^(p+1) - (x-1)^(p+1) = (p+1)x^p + g(x). Why don't we just subtract a polynomial from x^(p+1) so that the g(x) part would be canceled out. In other words, subtract a polynomial which would be the summation of g(x) evaluated at 1,...,n. We can do this inductively and then we're set. I've typed up an example of going through these computations in LaTeX and have put them up as an imgur image imgur.com/a/H650vOy And here is those two examples that you can verify the validity of in desmos. www.desmos.com/calculator/lgzjtzdjh8 Let me know if further clarification is needed.