How To Derive The Sum Formula of a Geometric Series - Free Educational videos for Students in K-12

How To Derive The Sum Formula of a Geometric Series - By The Organic Chemistry Tutor

Transcript


00:00	in this video , we're going to talk about how
00:02	to prove the formula that will help us to calculate
00:06	the sum of a geometric series and there's two of
00:09	them . There's a finite geometric series and the infinite
00:12	geometric series . We're going to talk about how to
00:14	prove the formula to calculate the sum of both of
00:17	those . So let's start with a geometric sequence .
00:22	Let's say we have a sequence three six 12 24
00:30	48 96 and so forth . three is the first
00:36	term . Six is the second term . 12 is
00:39	the third term . The common ratio is too you
00:46	need to multiply the first term by two to get
00:48	the second term three times to a 66 times two
00:52	is 12 , 12 times two is 24 . So
00:57	we have a common ratio of two . So no
01:01	to at the second term is the first term times
01:05	are six is 3 times two . The third term
01:11	is the first term times r squared 12 , His
01:17	three times 2 squared What three times for . Likewise
01:22	, if you want to find 1/4 term , it's
01:24	the first term times our cube 4 -1 History .
01:33	Make sure you understand that . Because we're going to
01:35	use that later to prove the formula . Now ,
01:39	what we have here is a sequence to convert it
01:43	into a series . We need to use the addition
01:46	sign . So this is a geometric series . So
01:54	let's add up the first six terms . So this
01:57	is going to be S6 , the partial some of
02:00	the first six terms , three plus six plus 12
02:07	plus 24 plus 48 plus 96 . That's 189 .
02:18	Now , let's use the formula to calculate that son
02:22	. This is the formula that helps us to calculate
02:24	the sum of a finite geometric series . It's a
02:28	sub one times one minus R a c n over
02:33	one minus R . So to calculate as sub six
02:37	is going to be a sub one the first term
02:39	which is three Times 1 -2 . Ours too .
02:45	It's race to the end And is basically the number
02:48	of terms which is six Over 1 -2 . So
02:53	this is going to be three one minus two races
02:57	. Six power . If you multiply 26 times You
03:01	get 64 , 1 - She was -1 , 1
03:06	-64 is -63 . The two negative signs will cancel
03:10	becoming positive . So it's three times 63 . three
03:15	times 6 is 18 , three times 3 is nine
03:18	. So we get 189 which is the same as
03:21	what we see here . So this is the formula
03:24	that helps us to calculate to some of a finite
03:29	geometric series . Now , what if we were to
03:34	have an infinite geometric series ? So let's say we
03:39	have the series eight plus four Plus two Plus 1
03:44	plus one half plus 1/4 and then plus dot dot
03:50	dot . So this series here it has the beginning
03:55	and it has an end . So it's a finite
03:58	geometric series . This series doesn't have an end ,
04:02	it goes on forever . So it's an infinite geometric
04:05	series . The common ratio , if you take the
04:10	second term divided by the first four divided by eight
04:13	is one half . The combination is 1/2 for all
04:16	terms . Now because the common ratio or rather the
04:21	absolute value of art Because it's less than one .
04:24	The series converges which means that the sum is finite
04:30	. If the absolute value of our if it was
04:32	greater than one , this series will diverge . And
04:35	you're you wouldn't be able to calculate the sum because
04:38	it can increase towards positive or negative infinity . So
04:42	it's important that the common ratio , the absolute value
04:45	of income ratio be less than one for this to
04:47	work . And the sum of this infinite geometric series
04:53	, we can write s sub infinity . It's the
04:58	first term divided by 1 -1 . So in this
05:01	example it's 8/1 minus a half . 1 -1 ,
05:09	eight divided by a half . If you multiply the
05:12	top and bottom by two , This becomes 16/1 ,
05:16	which is 16 . So if you were to add
05:19	these numbers eight plus four is 12 plus two ,
05:23	that's 14 plus 1 , 15 Plus 1 15.5 Plus
05:30	14 , If you keep doing this you're gonna get
05:33	closer to and closer to 16 but you're not going
05:35	to pass 16 . So that's the some of this
05:40	infinite geometric series . Now let's talk about how to
05:43	prove this formula as well as the other one .
05:50	So let's start with the formula that describes the sum
05:53	of an finance geometric series . So we're gonna have
06:00	S . S . N . Is a one plus
06:05	a two plus a three plus a four . And
06:13	then the second to last term is a sub N
06:15	-1 And the last term will be a seven .
06:24	Now , a sub two . Recall that a sub
06:27	two is a 71 times are and a sub three
06:32	. We said that it's a 71 times R squared
06:36	A four is a sub one times our Cube .
06:42	By the way , the formula that describes the f
06:45	term of a geometric series Or a geometric sequence is
06:50	a sub one . Our race to the N -1
06:55	. So in our first example where we had the
06:56	series or the sequence three , 12 24 48 let's
07:04	say . If we want to calculate the fifth term
07:06	, It would be a sub five Is equal to
07:09	a sub one which is three times are ours too
07:12	. Race the end -1 Or 5 -1 . 5
07:16	-1 is four . 2 to the 4th power two
07:20	times two times two times two at 16 , three
07:23	times 16 is 48 . So this formula here gives
07:27	you the f term of a geometric sequence . So
07:40	listening about that , if a sub N is a
07:43	sub one times are Race to the N -1 ,
07:47	What is a sub and -1 ? A 7 -1
07:52	. It's gonna be a sub one times are but
07:55	we're going to plug in and -1 into that expression
07:59	. So replace an end With N -1 . We
08:03	get a sub one . Are race to the End
08:06	monastery And that's what we're going to replace this thing
08:09	with . It's going to be a someone are To
08:14	the end -2 and then a sub N . We
08:17	already have that here . That's just a one To
08:21	the end -1 . Now for the next line ,
08:30	what we're gonna do is we're going to take this
08:32	equation and we're gonna multiply it by our so the
08:36	speed , this is gonna be our sn and then
08:41	this is going to be a one times are this
08:47	becomes if you multiply by our it's going to be
08:50	a one R squared and then plus a one .
08:54	Our cube plus a one Are to the 4th and
09:02	then multiply this by our The exponent is going to
09:05	increase by one . So N -2 plus one that's
09:11	going to be and minus one multiplying this by our
09:16	the export will increase by one as well . So
09:19	n minus one plus one Becomes simply and -1 and
09:23	positive one will cancel . Now what we're gonna do
09:28	is we're gonna multiply this equation by -1 . So
09:32	this is going to be negative . And then every
09:35	term within the bracket , once we distribute the negative
09:40	sign will be negative . So now we're gonna add
09:45	these two equations . So we're gonna have S seven
09:50	N plus negative R . S . F . N
09:54	. Or simply ss men minus R . S .
09:58	Event . And then we'll have a one . Now
10:07	A one AR -1 R . They're going to cancel
10:14	a one R squared minus a one R squared .
10:20	They will cancel as well . A one Our Cube
10:23	is going to cancel A one arch . The fourth
10:27	will cancel with something that's here and then A one
10:35	RN -2 will cancel with something that's there . And
10:39	then a one RN -1 will cancel as well .
10:45	This one will not cancel so but it has a
10:47	negative sign . So it's gonna be negative . A
10:51	one are raised to end . Now what we're gonna
10:54	do is we're going to factor out s event S
10:59	. M.N . by itself is one negative R .
11:03	S . F . N . Divided by S .
11:04	F . N . Is going to be negative .
11:06	Are Here we're going to factor out the first term
11:09	a sub one . A someone divided by itself is
11:13	one negative A one R . C N divided by
11:17	a one is negative . Our race to the end
11:22	And then we're gonna divide by 1 -2 . So
11:29	this gives us the formula for the partial sum of
11:31	a finite geometric series . It's the first term times
11:36	one minus the common ratio race to the end over
11:39	one minus R . So that's how we can derive
11:45	the son of a finite geometric series . Now ,
11:50	let's talk about how we can get this formula in
11:56	order to get that formula , we need to realize
11:59	that if the the absolute value of our is less
12:04	than one , and as N goes to infinity ,
12:10	our race to the end Goes to an Eagles0 ,
12:14	not infinity . Now , let's talk about that .
12:18	So if we have .9 , which is less than
12:21	one , raise the first power , that's 0.9 .
12:25	But what happens if we increase the exponent from 1-10
12:30	? Will this number get bigger or lower .9 ?
12:35	Race to the 10th Power Is a smaller number .
12:39	It's .348 678 Let's use the approximate symbol . Now
12:53	, let's increase it to 100 .9 . Race to
12:58	100 . It's even smaller , it's point 00002656 .
13:09	So we can see that the limit as N goes
13:15	to infinity of our sub . And assuming that our
13:20	is less than one or the absolute value of our
13:22	is less than one . This Becomes a zero when
13:27	an approaches infinity . So the sum of an infinite
13:35	geometric series where The absolute value of ours less than
13:39	one . This is going to be the limit as
13:44	N goes to infinity of the the finite geometric series
13:48	formula , which is a sub one times one minus
13:52	our race of the N Over 1 -2 . Now
13:57	this is the only part of the equation that has
13:59	ended And we know that are to the end will
14:02	go to zero once we apply This limit , so
14:06	it becomes a sub one times 1 -0/1 -2 .
14:11	And 1 0 is simply one . A sub one
14:17	times 1 is simply a sub one . So the
14:20	sum of an infinite geometric series Boy , the athlete
14:24	valley of ours , less than one Is this equation
14:27	a sub 1/1 -2 . So that's how you could
14:31	derive it from this formula . It's by realizing that
14:36	as N goes to infinity are certain goes to zero
00:0-1	.

Summarizer

DESCRIPTION:

OVERVIEW:

How To Derive The Sum Formula of a Geometric Series is a free educational video by The Organic Chemistry Tutor.

This page not only allows students and teachers view How To Derive The Sum Formula of a Geometric Series videos but also find engaging Sample Questions, Apps, Pins, Worksheets, Books related to the following topics.