Finite geometric series formula justification | High School Math | Khan Academy

[@muhammadericali]

I just graduated with a degree in Biology and you were a huge contributor to that! Thank you.

[@pistonx3544]

How are you doing now?

[@ian.ambrose]

@@pistonx3544 He is doing boxing now.

[@humzahkhan6299]

This is absolutely awesome ! Thank you 🙏 Sal 😃😁. For those that are confused, all he did was take any geometric series called Sn, subtracted that same series Sn but also multiplied by a factor r (subtracting a positive and adding a negative get the same result, this also works if u do Sn * r - Sn, u just get a different formula that also works). This little trick leaves u with (Sn * r) - Sn equaling a - ar^n (from the video). Finally, since this is Sn - Sn*r it’s also just the pattern Sn multiplied by (1-r). If u know the pattern of Sn u also know ‘a’, ‘r’, ‘n’ meaning u can put in every necessary variable into the equation and solve for Sn.

[@SS-bv3xh]

Nicely explained. Thanks a lot.

[@曰_曰]

what a beautiful formula

[@AlexeyYunoshev]

Thank you very much. It's beautiful.

[@GOODBOY-vt1cf]

thank you so much

[@huzaifaabedeen7119]

why multiply with -r?

[@paragshahusg7351]

great explaination

[@shamim-io]

Thank you so much sir... Greetings from india!!

[@yeonhojung7185]

thank you so much!

[@RaghavendraLingayya]

Dear Sir, The formula derived by multiplying (1-r) , i.e., denominator part of the final formula. If series formula is of format, Question (Q) = Numerator(N) / Denominator(D) , i.e., Q = N/D, ----(1) then Q*D = N , Substitute the value of "N" in (1) we get so Q = Q*D /D . Using this logic, the formula is derived. Means, if we know denominator part of the formula, then we can find the the formula , i.e., numerator part. For example, in sum of GP formula, D = (r-1) , Q = (Sum of GP) So, Q*D = (a +ar + ar^2+ up to n terms) * (r-1) = numerator part of answer = a(r^n-1) So, Q = Q*(r-1)/(r-1) = a(r^n-1)/(r-1) Derivation should not start knowing the answer before. Hence, this method is not derivation. So, stop misguding students. Its a big offence. Tell the truth for the development of easy and quality education. Here, my question is how you come to know that multiplying (1-r) or (r-1) or doing it in two steps like multiply -r ...?

[@yeonhojung7185]

dear sir, prove any other formula without that then. the explanation is *not* derived from the formula, but *the explanation* itself derived the formula. Not that you know the formula, you can't be exclaiming _"Ahhh! This formula derived the whole explanation!"_.

[@yeonhojung7185]

also, every textbooks describes it this way too :-)

[@zouraizkhan2467]

what would happen if he had used positive r times S sub n ?

[@piratesgaming3738]

sir can you solve this problem For -1 < r < 1, let S(r) denote the sum of the geometric progression 43 + 43r + 43r2 + 43r3 + …….. Let ‘a’ between -1 and 1 satisfy S(a). S(-a) = 2021. Find the value of S(a) + S(-a).

[@michieldrost9396]

is it corrent of me to tihnk of the series as; Sn = ar^0 + ar^1 + ar^2... + ar^(n-1) + ar^n instead of; Sn = a + ar + ar^2... + ar^(n-1) + ar^n Using the exponents helps me remember the series better. Merry Christmas.

[@FTWTitanium]

+Michiel Drost Yes, why not. Any number on power 0 is 1. So, a*r^0 = a*1 = a.

[@LearnMathsFree]

+Michiel Drost As the previous poster said, that's perfectly fine. r^0 and r^1 are usually replaced with 1 and r, however.

[@shakuntalajain9788]

Michiel Drost i remember it in the same way😁

[@LearnMathsFree]

Thanks for the video.

[@mostafakhaled9702]

Hey Khan can you make a video proving that the medians of a triangle are concurrent? Or is there a video already?

[@nadirajames86]

i still need more to learn more information

[@nadirajames86]

+Nadira James 😍😍😍😍😍😍😍🐩🐩😂i can't believe it

[@sapiently1]

My question is why multiplying it simplifies it to a form that you can convert back to get your answer. I just don't understand why this simplification works. I get that it's multiplying (1-r) to the whole sequence to leave the whole entire middle part and leave only the first and last term, so you can divide it back to revert it to the original sequence so you get your answer. I just don't understand why this simplification method works.

[@tseptengurung1179]

same brother. there are so many things that i can see HOW it works, but i don't understand WHY it works

[@SpiritVector]

I find this very strange because some guy would of had to come up with the idea to muliply an entire series by some negative r and then subtract it from the original series to get a formula that sums the original series up. Super cool though!

[@matthewkilgore3430]

You don't need to multiply by negative r, all you have to do is extract positive r from all the terms except the first, and realize what remains inside the parentheses is S_n - a*r^(n-1). The proof flows more naturally that way in my opinion as it doesn't involve any massive term cancellations. I think you can find this alternate proof online but I can make a short vid showing you the steps if you want.

[@SpiritVector]

@@matthewkilgore3430 Its been awhile but I do understand it.

[@SpiritVector]

@matthewkilgore3430 actually, what would be awesome if you knew anything about it would be explaining the field of view transformation equation.

[@matthewkilgore3430]

@@SpiritVector Sorry, I'm not familiar with that equation.

[@DexkillTutorials]

ar³, im a pirate

[@pixelgames5566]

Second

[@zachpejeau3584]

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