Sequences and Series (part 1)

[@Trevtutor]

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

[@PressStartLetsPlay]

***** I don't think that is true. I have a video from 2009 on my old channel and the quality is at 480p.

[@sehrgrossesglied5000]

funny that this might be the aquivalent for young kids to what for us is bad quality oldschool cable TV :D

[@tacitusignotus6201]

To be honest the quality KZbin is today is going to seem sad to kids 10 years from now.

[@Maryam-go3yq]

The fact that this comment is 4 years old messed me up

[@TheCoward383]

Actually not that bad

[@Nikifuj908]

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,..."

[@AdrianLara-rr1ow]

i heard difficult :I

[@mjl7810]

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

[@crazycaramel39]

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.

[@gundream8384]

Khan is a good teacher. My teacher is a caltech Phd, but it took him 20 minutes to explain this concept.

[@jecht86]

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.

[@dbmasters]

damn this was before Sal was a premier partner - he still had the 10 min video limit... excellent as always, mate

[@JasonPoh89]

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

[@khanacademy]

You're right. I fix them in the next video.

[@rj-yw4mu]

If youre watching this in 2021 YOU ROCK

[@anticemlyne]

This is amazing!! It all makes sense now. I wish my professor would watch these.

[@mayankbhatt7241]

(N-1), N NOT N-2

[@dhruvverma6938]

Mayank Bhatt you get the point.

[@CaptainSuspenders]

This was a life saver. I go to an online school, and the way they teach math is a little.. iffy. It's very difficult to learn, and I was starting to lose faith in my own intellect. Thank you for restoring it. :]

[@LikidySplitz]

If this is too presumptious, God help you. Thank you khan, you saved my life again

[@Nickuncle]

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

[@Raiyun166]

Thank you so much for expaining. My teacher just randomly started writing these formulas without saying anything and everyone was just sitting there. We could work out what he asked us by following his examples and replacing numbers but we had no idea what the reasoning behind it was. Heck, I didn't even know it was series and sequences.... >.> (and trust me, I was paying attention real hard, and other people in my class couldn't explain it either and said they needed to consult their text book)

[@PianoWallaby]

This one belongs on your greatest hits.

[@brokrishnan]

Thanks so much for this! It really helped me understand series. I'm pretty sure my Precalc teacher speaks another language.

[@flip2shoot567]

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

[@hourglass1435]

me too. that unjustifiable b+ am i right?

[@SmokeDankTrees]

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

[@andrewp3646]

+Patrick G Carl Friedriche Gauss came up with that representation in elementary school lol, what a freak of nature he was.

[@karunanidhi9463]

Thank you sir and your teaching so amazing

[@ebercota5652]

khan, you are honestly amazing

[@BryanBryce]

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

[@ronanmacfhlannchadha7774]

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

[@afapefreak546]

We need more of this guy

[@thetimbo21]

Sal, you are THE MAN.

[@DTNRom88]

N-1 being the last point before N. For example if you're counting up to 5 you would go 1,2,3,4,5. if you replace 5 by "N", then what would be the theoretical value of 4? It's N-1 (5-1 = 4) Basically N-1 is just a way to say the before last number in your sequence

[@carlosav]

This is a great trick, but it only works if you're taking the sum of a sequence of positive integers with one as its starting point. There is a more general formula for finding the sum of all the integers between any two positive integers a and b (b > a) : (a+b)(b-a+1)/2

[@jhonma1]

Thank you Khan.

[@SaiF290]

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

[@AVGstrEnjoyer]

thank you homie..your videos are great

[@alexPsanz]

im not done but im getting the feelin my mind is about to get blown.

[@julienalonso22]

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

[@Nandine2]

khan is king!!!

[@aS0yCz8H7Fy4SAG]

thanks for the proof i've always been wondering why the sums of N is n(n+1) / 2

[@TheFrycry]

Graet video! sal thanks!

[@Vengetic]

@InnocentxR He's just giving it as an example, instead of just going to N say for example N is 100, he's saying N-1 + N which will be 99 + 100 he can do this because the sequence is increasing by 1 everytime.

[@shersher1492brewer]

The burb at 7:07 is my favorite.

[@callumvanheerden1530]

Isn't this basically like for loops in programming?

[@keyser1989]

Deft ZA exactly!!!

[@callumvanheerden1530]

Wow! first time programming stuff is helping me in math class!!

[@slaanesh_pet]

That's exactly the premise!

[@anodaone6525]

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)

[@Alejandro-Te]

However, he mistook. The last number should be to the power of "n", obviously.

[@yn2464]

on part 2 he changes it

[@Alejandro-Te]

yun kim Thanks for the answer. Don't tell me you're the guy in the picture.

[@vivekanandan5591]

Really Gud

[@hellocinnamon]

Wow!! This is amazing. Thank you!

[@13430]

I might not be the only one to watch this after 12 years time flies ;))

[@rahulmehta7476]

@@13430 Time flies too fast

[@GoodVolition]

A whole SERIES of fields.

[@breeishly]

+Cameron Goode I was so excited to point that out that pun but I scrolled down and this was the first comment I saw... damn it.

[@user-pw8qc9nd9k]

You pun thirsty cockroach

[@salmankhan-wp6vo]

you saved my life bro .

[@madhavkhindri2817]

The final expression is only valid if first term is 1 and common difference is also 1, the general expression is the sum of an arthemetic expression i.e s = n/2*[2a + (n-1)*d]

[@anshajkumar6542]

I just found a old hidden treasure😀😀

[@wetim99]

n is the last number in the series you are adding up, so the one before it would be n-1, two before it would be n-2, etc.

[@nehaagrawal6680]

nice.....

[@nahcrew05]

Very good bro

[@SockMonkeyRacer]

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?

[@RagzDoll94]

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.

[@tuckeyalanaj]

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

[@THEFOUNDER865]

You should go more general. Arithmetic isn't just +1, you can also +4, +5....etc. Aka you can sum up n terms by adding the same amount of any number, not just 1. So the general formula is n/2(2a+(n-1)d) and what does 2a+(n-1)d=? Well, it equals the first term (a) and the last term (a+(n-1)d). So another form it can take is n/2(a+l) where a is the first term, and l is the last.

[@dbmasters]

@ACfireandiceDC There's always the fundamental basics.

[@naughtydevil02]

that is so awesome!!!:)

[@rulerofmanandape]

You just made sense of sigma notation for me. Burp all you want. :)

[@MisterTutor2010]

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

[@StephenLee1]

hey Sal, can you do some vids on the other convergence tests in depth. Integral test, telescoping series, nth term root test, lim comparison test etc. thanks.

[@hollowmoon7636]

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

[@jemappellelanoosh1743]

Hahahha ! that burp part made me laught :-) and once again thank you Khan ...

[@jingle552]

thanks for this video. I was wondering how did the formula come about... all the while I was applying a meaningless formula that my teacher said that we have to memorize.

[@hedonism13]

Now I'm a magician! @sajidullah Buzzkill of my excitement.

[@KeyOfAm]

@crodd63 Does make sense, I dont really get why it goes a^n-1 , a^n-2

[@wajeeh.hassan]

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

[@enjoyablesounds]

As of last week I started studying "sequences and series". Again, the branding chosen inappropriate and unnecessarily confusing , not distinguishable at all. I would leave "sequences" as it is but would refer to "series" as "add-series" or better yet "add-sequence or sum-sequence" simply because now, in the word the clue of summation already has been giving in the initial descriptive sense. what do you think?

[@david887]

No, I thought that too, but when I went back to it I realized that he actually said "The most typical sequence..."

[@candacevalenzuela9237]

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

[@VSPG_SIVANI]

At 5:26, What am I doing? LOL

[@Akrubix]

@Thymonico I think he say's "typical", not "difficult". :)

[@iwuvvparamore]

Hi, I tried using this formula for calculating the sum of an arithmetic sequence that did NOT have a common difference of 1 and the answer was incorrect. Why would that be ? Should we be using the formula Sn = n/2 [2a + ( n -1 ) x d ] instead as it gives you the correct answer for the sums of series with a differene that is not 1 ? Thanks

[@hammurambi]

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

[@navidhasan236]

Thanks.

[@Wandelaarke]

Thank you :)

[@RediatBekele]

Nice explanation but, the geometric serious should not go up to N, rather it should go up to N-1 as it starts with k=0. the formula will not be right if we are using it with N+1 as the power on the first term.

[@kamrannasir3871]

ya, he corrects it in the next video..

[@onlyAerik]

Why is these 2 videos listed after the solid of revolution videos in the calculus section? I would've put these at the end of the precalculus series so we can have a better intution about summing infinite numbers of infinitely small changes (dx)

[@_A.C.C.]

wait, the [n(n+1)]/2 is only if the number starts at 1 and increases by 1?

[@kisasohma9]

@FlyWithMe4ever Are you referring to nigahiga? The awkward moment when you say goodbye but you walk in the same direction. That has totally happened to me before, lol. Thumbs if you either watch nigahiga videos or had that happen before-where you walk in the same direction after saying good bye

[@mjl7810]

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

[@tedoymisojos]

I didnt understood how you got at th eformula at all:S but I get how to use it

[@scikud]

hey i have a question~ why are we adding the sequence in reverse.. i mean i understand its function but why intuitively would we add this reverse sequence to the original

[@lucile1010]

@InnocentxR I think, I think he's trying to show us where d formulae come from?x

[@kamyarghofrani]

cuz they are in order, if the last number is N, the second last one is N-1

[@alexeybaryshev1947]

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

[@yourfavouriteusername]

How'd your IB go fam? Did this video help? I JUST started my first year of IB. Funny how time works, huh? Please let me know!

[@deepmalakumari3425]

Nyc vedio

[@LadyHelenaRavenclaw]

does anyone know if he's done videos for the divergence/convergence of a series??

[@Max11551]

I think at the end of the video there is a mistake. You say the last elements of the series are a^(N-1) and a^(N-2), but shouldn't that last one have been a^N ?

[@peachy1309]

woah.. I don't understand :(

[@crismal6477]

Do you understand it now?

[@htie96]

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

[@rinimukherjee]

7:02 best thing ever

[@Hazit90]

i got a like for the burp.lol

[@Alejandro-Te]

Am I mistaken or, since k=,1 the formula should be n*(n+k)/2? Otherwise the formula doesn't apply to other sequences.

[@miif2316]

Citriano Torres --- No, it's strictly (N(N+1))/2. You, yourself, have stated that k=1, so it can't be equal to other values, and I have proven your point to be wrong myself. For example, take k=5 and N=6, and if we were to apply the formula S=(N(N+k))/2, then we would get 33. However, if we were to use the sigma notation, then we would obviously get 11, which is 5+6.

[@acm22fan]

how is this guy so smart?

[@rayz3678]

This for core 1 a-level?

[@KatieLizzy]

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

[@PR3D470R18]

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 ._.

[@PR3D470R18]

Oh well its fixed in the start of part 2 sorry xD

[@mrpersonwithaname]

@crodd63 he corrects that in part 2

[@corbinsherman2438]

you learned series in precalc?

[@Hellsslave666]

@abombinreverse Do you need glases?

[@crodd63]

i do think that at the end of the video, (Geometric) it's not supposed to be a^(n-1) + a^(n-2), i think its supposed to be a^(n-1) + a^(n)

[@josefinej3921]

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