Let’s learn a little bit

about sequences and series. So what’s a sequence? Well, a sequence is just a

bunch of numbers in some order. You know, the most difficult

sequence is 1, 2, 3, 4. You get the point. And what’s a series? Well, it’s often represented–

it’s just a sum of sequences, a sum of a sequence. So, for example, the arithmetic

sequence– sorry the arithmetic series is just the sum of

the arithmetic sequence. So 1 plus 2 plus 3 plus–

we could keep going until maybe some number. This is called the

arithmetic series. Nothing too fancy here. But before we move forward,

let’s get a notation for how we can represent these sums

without necessarily having to write out all of the digits or

having to keep doing this dot, dot, dot, plus notation. And that notation

is Sigma notation. That’s an upper case Sigma. And how do you use

Sigma notation? Well, let’s say I wanted

to represent this arithmetic series. So I would say, well,

let’s add up a bunch of– let’s call them k’s. This is an arbitrary variable. And we’ll start at k equals 1. We’ll start at k equals

1, and we’ll go to k is equal to big N. And this is the exact same

thing, so we first make k equal to 1, and then we add it to k

is equal to 2 plus 3, and we go all the way until N minus

1 and then plus N. So this is the Sigma notation

for the arithmetic series. Before I move on, I think this

is a good time just to like learn a little bit more about

the arithmetic series. We’ll actually focus on this

one and the geometric series because those are the two

that you’ll see most often. And then once you learn

calculus, I’ll show you the power and Taylor series, which

is exact– the Taylor series is a specific version

of a power series. But let’s play around with this

arith– I keep wanting to say arith-MET-ic, but a-RITH-metic,

either way– series. So let’s call the sum S. Let’s say that this is equal to

the sum from k is equal to 1 to N of k, which is equal to, just

like we said, 1 plus 2 plus 3. And we’ll just keep adding

them, dot, dot, dot, to a bunch of numbers, to big N minus

1 plus big N, right? Fair enough. Now, bear with me a second. I’m just going to write that

same exact sum again, but I’m just going to write

it in reverse order. And I think it’s intuitive to

you that it doesn’t matter what order I add up numbers in. They’ll add up to

the same number. 2 plus 1 is the same thing

as 1 plus 2, right? So let me write this exact

same sum, but I’ll write it in reverse order. So that’s the same thing as N

plus N minus 1, plus N minus 2, plus– and the pluses

keep going– plus 2 plus 1. This is the exact sum, just

in the reverse order. And I did that for a reason

because now I’m going to add both sides of this equation. I’m going to take– S plus S. Well, that’s just 2S. And that’s going to equal

this sum plus this sum. I wrote this so that

the sum becomes clean. And why do I say that? Well, let’s add up

corresponding terms. We could have added up any

terms, but– so since they all have to add up, let’s just add

the 1 plus the N, then we’ll add the 2 plus the N minus 1,

then we’ll add the 3 plus the N minus 2, and so forth. And I think you’ll see in a

second, or maybe you already realize why I’m doing this. One plus the N, the 2 plus the

N minus 1, the 3 and the N minus 2, all the way to the N

minus 1 and the 2, the N and the 1. What’s 1 plus N? Well, that’s just

N plus 1, right? What’s 2 plus N minus 1? Well, that’s also

N plus 1, right? What’s 3 plus N minus 2? I think you could guess. It’s N minus 1. And we just keep doing that. And what’s N minus 2 plus 2? Sorry, this is a plus. N plus 1. And what’s N plus 1? Well, that’s just N

plus 1, of course. So my question to you

is how many of these N plus 1’s are there? Well, there are N

of them, right? Each N plus 1 corresponds

to each of these terms, so there are N of these. So instead of just adding N

plus 1 N times, we could say that this is just

N times N plus 1. So we have 2 times the sum is

equal to N times N plus 1, and we could divide both sides by

2, and we get the sum is equal to N times N plus 1 over 2. Now, why is this neat, or

why is this cool at all? Well, first of all, we found

out a way to sum this Sigma notation up. We got kind of a

well-defined formula. And what makes this especially

cool is you can use this for low-end parlor tricks. What do I mean by that? Well, you can go up to someone

and you can say, well, how quickly do you think I can add

up the numbers between 1 and– what am I doing– oh,

between 1 and 100? And, you know, people will say,

oh, it will take you a little time: 1 plus 2 plus 3. And you say, well, it takes

me no time at all because this is what I can do. So the sum– and I just want

to show you that you can use different variables from B

equals 1, we’re taking the variable B, to 100, right? That’s the sum from 1 to 100. And we figured out

what that formula is. It’s going to be 100

times 101 over 2. Well, what’s 100 times 101? It’s just going to be 101

with two zeroes, right? 10,100 over 2, and

that equals 5,050. That’s pretty neat. Instead of having to say 1 plus

2 plus 3 plus blah, blah, blah, blah, blah, blah, blah, blah,

plus 98 plus 99 plus 100, this would take you some time, and

there’s a very good chance you would make a careless mistake. We could just plug into this

formula, which we proved and hopefully you understood,

and say that equals 5,050. You could do even something

more impressive: the sum from 1 to 1,000. What’s the sum from 1 to 1,000? Well, our formula, remember,

was N times N plus 1 over 2. So if N is equal to 1,000,

then what’s our sum? It’s 1,000 times 1,001 over 2,

which is equal to– well, we’ll just add three zeroes to

this: 1,001, one, two, three. Sorry, I think that was

my first burp ever on one of these videos. I should re-record it, but

I’m going to move forward. That kind of disconcerted

me a little bit. I’d eaten too much. Anyway, divided by 2,

and what is that? Let’s see, it’ll be 500–

let’s see, this is a million. Half of a million is 500,000. 500,500. And that would have taken

you forever to do manually. But based on this formula we

just got, you know how to do it very, very quickly. So that’s the

arithmetic series. But let’s do another one. This is another typical

series that you might see. Actually, this one you’ll see a

lot in your life, especially if you go into finance or really a

whole series of scientific– this shows up a lot, and this

is called the geometric series. And the geometric series is–

essentially you take x. And I’ll do it generally where

I just take a variable x, and I say– well, no, no. Let me just not take an x. Let me just take some number. So let’s say some number a to

the k from– I don’t know. Let’s say from k is equal

to 0 to k is equal to N. What does that mean? Well, that means a to the 0,

right, k is 0, plus a 1 plus a squared plus a to the third

plus– and you could keep going– plus a to the N minus

1 plus a to the N minus 2. This is called the

geometric series. And it might not be obvious to

you, but this type of growth, where you keep increasing the

exponent, this is called geometric growth. So how do you take

the sum of this? Well, let’s see if we can do a

similar trick, although this trick will involve

one more step. So let’s call the sum S. Let’s call it the sum from k

equals 0 to N, a to the k. And that, of course, is

equal to what I just wrote. I probably didn’t have

to do it like this. a squared plus bup, bup, bup,

bup, plus a to the N minus 1, plus a to the N minus 2. Now let’s define another sum,

and I’m going to call that aS. Actually, I’m about to run out

of time, so I’ll continue this in the next video.

(n(n+1))/2

My teachers are really good but this is really good on the side in prep for Math team.

your math, great, your parlor tricks, not so much…

I should just not go to math class.

I love how he says "The most typical sequence is 1,2,3,4,…" and the subtitles say "The most difficult sequence is 1,2,3,4,…"

Sorry to hear about all of the mediocre teachers. Fortunately for me, my teacher is awesome!

Hahahha ! that burp part made me laught ๐ and once again thank you Khan …

HA! A whole series XD

at the end it shouldn't be a^(N-2) should just be a^N

1st time i see a WHITE BOARD !!! NOOOOOOOO

um at 9:03, it should be A raised to power N. you wrote A raised to power N-2. that should come before A raised to power N – 1. just saying.

In contrast to (well, I think) the majority of people here, my teacher isn't underqualified. It's just that she overcomplicates EVERYTHING until our heads are spinning.

This actually makes so much sense.

khan. You are the definition of a pimp.

Why is a^n-1 written before a^n-2 in geometric sequence?

I absolutely love your videos, and you save my life when time comes for my calc final, but I can't lie: that burp moment was my favorite part out of all of these videos hahaha.

Private enterprise beats government any day. The power of the individual is amazing.

So cool! My maths teacher never told us how the formula came about! ๐ For arithmetic series.. this is so useful…

Hah, Gauss discovered a formula when he was in the first grade of elementary school ๐

um he clarifies that in part 2 of this video. just saying. ๐

And what, they should be run by the ever-objective, omnipotent, and totally fair government? You make it seem as though the a business or a government isn't already a group of people who do in fact make money from their jobs.

The only difference if it were privatized would be that competition would drive up quality, and drive down costs; and there would be significantly less wasted funds. Private schools spend less per student and provide far better education.

I'm not for privatizing prisons, let me make that clear. However, the administration of the prison doesn't decide whether or not people are arrested. That's up to the cops — if we want to reduce our prison population, we should legalize drugs, stop targeting minorities, and most of all, stop judging a cop's performance by his/ her number of arrests, and rather by the quality of their work.

How did this shift from sequences and series to privatized jails?

I was curious as to what you were doing until I realized that was a formula for sums! What! Math.

Sal, you are THE MAN.

Why is it a^( n-2 ) in the geometric series…?

legalize drugs? are you serious

why except for the minorities part? care to explain?

legalizing drugs, namely marijuana, would actually solve a few issues. one of the main reasons weed hasn't yet been legalized is the high profit made from drug busts. the police departments and the government in general make a shit ton of money off of marijuana being illegal. and more money for the government is what the government wants. ergo, illegal weed. c:

What in the hell is going on in the comment section? God damnit youtube.

Kid crying @5:53. Thanks Sal for your personal sacrifice to teach.

My teacher pronounces it air-ith-medic and it drives me insane

Sum of all number between m and n = (n^2-m^2+n+m)/2

If m =1 then it reduces to (n^2+n)/2

I'm pretty sure that's how the adjective form of "arithmetic" is pronounced.

sigh.. all you had to do was watch part two.. he said that was a mistake..

7:02 best thing ever

you saved my life bro .

Yyy

Poda fuel

That kinda disconserted me a little….I- I eat too much.

That kinda disconserted me a little….I- I eat too much.

You gotta give the ending a speech bubble to correct the mistake you made (I think), the last term should be a^n, not a^(n-2). Correct me if I'm wrong ._.

This might as well be an ASMR vieo.

i got a like for the burp.lol

Very useful lesson! Only, please get a tablet….

wait wait… what???

why is the quality so poo?

Did he just burp? I wish my math teachers would burp.

this is beneficial however n(n+1)/2 will only work if the numbers are incrementing by 1, if the start going up in 2's or 3's this will now work, you will have to use n/2(2a +(n-1)d) where a is the start and d is the commen difference

To think all of YouTube used to be this quality. Crazy times.

1n a lottery, the first ticket drawn wins a prize of $25. Each ticket after that receives a prize that is twice the value of the preceding prize.

ย

a) Write a function to model the total amount of prize money given away.

b) How many prizes are given out if the total amount of prize money is approximately $2 million?

Can someone help me with this question?

(N-1), N

NOT N-2

What an elegant way of representing a series. Thank you.

at the end of the video, when adding the geometric series, a to n-2 comes before a to the n-1.That makes it more understandable because a to the n-2 is smaller than a to the n-1.Nice video though (y)

Thanks.

Am I mistaken or, since k=,1 the formula should be n*(n+k)/2?

Otherwise the formula doesn't apply to other sequences.

this does not help for the CFAT exam in your compilation of videos

My teachers cant really teach this stuff but this is just amazing, i have a B in math now

Riveting!

Really Gud

nice…..

I am so confused

wooow

I'm watching this before we even learn it to be prepared and… I'm scared to even start the video… O.O

internally crying1..1…2…3..5…10..15…25…42…next?

A whole SERIES of fields.

1st year IB Math exam in 5 days, starting topic one.. God help me

Why have u written (N-1),(N-2). in geometric series???

I don't think it's right!!!!

It must be (N-1),N.

Isn't it??

The pronunciation of "arithmetic" should be "ar-ith-met-ik" with MET stressed because the word is an adjective in this case.

woah.. I don't understand ๐

the encircle looks like a dic* just sayin

All this guy wants to do is confuse us. Itโs disgusting.

Isn't this basically like for loops in programming?

Why is it a^n-2 rather than just a^n isn't the exponent approaching n

PLEASE USE A SLIGHTLY BRIGHTER COLOR THAN SOLID BLACK!

It's supposed to be N (after the N-1) instead of N-2 !!

How you reach a(n-2) at last, when it should be a(n)…??

4:44 how is there "n" of "(n+1)"? wouldn't that mean n is infinity also meaning the n+1 = infinity+1 ?

Very nice video. Plz do watch my video on the same topic. Link is here. https://youtu.be/oVsp6D-wP84

there is an error in his video, the final term of your last series should be a^(N), not a^(N-2)

Very good bro

This tutorial doesn't make any sense at all..

why does geometric series go to n-1, n-2???

Nyc vedio

lmao the burp

I AM CONFUSIONI don't understand….. I hate math!!!!!!

im still confused

We can't describe how much we appreciate you uncle Khan

Although I think you made a mistake at 9:00 a^(n-1) then a^(n) not a^(n-2)

20 years after my 7th grade math instructor asked us this question, i run into his solution with my Calc II work. Looking back, that guy is optimistic about 7th graders being able to figure out the Sigma property.

similar to A.P.

Why the quality of the videos is so bad?

khan, you are honestly amazing

actually the most confusing math i have ever done. anyone else understand the math but not the mathmatical terms? and why n-1? where does that crap come from???

3:59 is it just me or is he drawing penises?

You suck at teaching

I can't understand โน๏ธwhy math is very difficult like my gf she is very difficult to understand that's why we break.

Khan academy came so far lol they became so HD but the content is still the same

I just found a old hidden treasure๐๐

someone explain what went through his mind when he decided to add the 2 series. like i get the end reason but how would he have known to add the 2?