Geometric Series and Geometric Sequences - Basic Introduction - Free Educational videos for Students in K-12

Geometric Series and Geometric Sequences - Basic Introduction - By The Organic Chemistry Tutor

Transcript


00:0-1	in this video , we're going to focus on geometric
00:02	sequences and series . So first let's discuss the difference
00:07	between a geometric sequence and a geometric series . What
00:14	do you think ? The differences ? Here's an example
00:19	of a geometric sequence , The # 36 , 12
00:25	24 48 and so forth . A geometric sequence is
00:31	different from And the different sects sequence such as this
00:37	one here . In that a geometric sequence has a
00:40	common ratio versus a common difference . If you take
00:45	the second term and divided by the first term six
00:49	divided by three is too , you're gonna get the
00:51	common ratio . If you take the third-term divided by
00:56	the second term , you'll get the same common ratio
00:58	. 12 divided by six . It's too . So
01:02	that's the defining mark of a geometric sequence . In
01:05	an arithmetic sequence , there's a common difference . If
01:09	you take the second term and subtract it from the
01:12	first term 8 -5 is dream . If you take
01:16	the third term subtracted by the second , 11 -8
01:19	Extreme . So that's how you can distinguish an arithmetic
01:22	sequence from a geometric sequence . And arithmetic sequence has
01:27	a common difference between terms A geometric sequence has a
01:31	common ratio between terms within an arithmetic sequence . You're
01:35	dealing with addition and subtraction for a geometric sequence you're
01:40	dealing with multiplication and division between terms . So now
01:47	that we know what the geometric sequences and how to
01:50	distinguish it from an arithmetic sequence . What is the
01:54	geometric series ? A geometric series is basically the some
01:58	of the numbers in a geometric sequence . So three
02:02	plus 6 plus 12 plus 24 and so forth would
02:08	be a geometric series . This is the first term
02:14	, this is the second term , this is the
02:16	third and so forth . Now the former you need
02:23	to calculate the f term of a geometric sequence or
02:27	series . It's the first term a sub one Times
02:32	The Common Ratio . Our Race to the N -1
02:35	. So for instance , Let's just make a note
02:42	that R . is equal to two . Let's say
02:47	we want to find the value of the fifth term
02:49	. We know the fifth term is 48 . But
02:51	let's go ahead and calculate . So you can see
02:54	how this formula works . So the first term is
02:57	three . Are the common ratio is too and n
03:02	is the sub script here . We're looking for the
03:03	fifth term soon and it's five . So 5 -1
03:09	is four , two to the fourth power . If
03:13	you multiply 24 times two times two times two times
03:16	two . That's 16 . 16 Times Street is 48
03:22	. So that's the function of this formula . It
03:25	gives you the value of the F . Term .
03:28	So you can find the value of the eighth term
03:30	, the 20th term and so forth . The next
03:35	equation you need to be familiar with 1st . Let's
03:38	get rid of this . The next equation is the
03:47	partial sum formula . The partial son of a geometric
03:52	series is the first term times 1 - Our race
03:56	to the end over one minus R . So let's
04:01	say that we want to find the some of the
04:03	first five terms . This is going to be three
04:07	plus six plus 12 plus 24 plus 48 . Go
04:15	ahead and plug that into a calculator . So for
04:23	the first five terms I got the partial summers being
04:26	93 . Now let's confirm that with this equation .
04:33	So let's calculate S Sub five . The first chemistry
04:38	Times 1 - R . RS two and it's five
04:44	Divided by 1 - are . So that's 1 -2
04:51	. 2 to the 5th power . That's 32 .
04:55	1 -2 is -1 . Now 1 -32 is negative
05:01	. 31 Three times negative . 31 that's -93 but
05:07	divided by -1 . That becomes positive 93 . So
05:13	we get the same answer . So any time you
05:15	need to find the son of a finite series ,
05:21	you could use this formula . So this series here
05:27	is finite . We're looking for the some of the
05:30	first five terms . There's a beginning and there's an
05:33	end . This series here is not finite . It's
05:40	an infinite geometric series . The reason being is because
05:44	of the dot dot dot that we see here it
05:46	goes on forever . It doesn't stop at the fifth
05:48	term . It keeps on going to infinity . So
05:50	it's an infinite geometric series . This is an infinite
05:57	geometric sequence . It's a sequence that goes on forever
06:01	and its geometric . So make sure you can identify
06:04	if a sequence is arithmetic geometric ? Is it finite
06:08	infinite ? Is it a sequence or series ? Now
06:25	the next thing we need to talk about is the
06:29	arithmetic mean And the geometric mean . Let's call the
06:35	arithmetic mean . Emma The arithmetic mean is simply the
06:39	average of two numbers . The geometric for me let's
06:43	call it M . G . Is the Square Root
06:47	of the Product of two Numbers . So let's go
06:51	back to the arithmetic sequence that we had here .
06:58	If we wanted to find the arithmetic mean between the
07:02	first and the third term , it will give us
07:05	the middle number , the second term . If you
07:08	average five and 11 and divide by two . Using
07:11	this formula You're gonna get 16/2 which is eight .
07:18	So the US when you find the arithmetic mean of
07:20	the first term and the third term you're going to
07:24	get the second term because the average of one in
07:26	three is too Yeah . Now let's find the right
07:35	protect me between the first and the fifth term .
07:38	This will give us The middle term 11 . So
07:47	if we were to add up a one and a
07:49	five and then divided by two . If we were
07:52	to get the average we will get a three ,
07:55	the average of one in five history . So let's
07:59	add five and 17 and then divided by two .
08:02	five plus 17 is 22 , 22 , divided by
08:06	two is 11 . So that's the concept of the
08:09	rhythm technique . Whenever you take the arithmetic mean of
08:14	two numbers within an arithmetic sequence , you get the
08:17	middle term of that of those two numbers that you
08:19	selected . Now the same is true for a geometric
08:22	sequence . If we were to find the geometric mean
08:25	between three and 12 , You'll get the Middle #
08:28	six . If we wanted to find the geometric mean
08:31	between three and 48 , We would get the Middle
08:34	# 12 . So let's confirm that . Let's find
08:38	the geometric mean Between a one and a three .
08:46	So the first term is three . The third term
08:49	is 12 , Three times 12 is 36 . The
08:54	square root of 36 is six . So we get
08:57	the middle number . Now let's find the geometric mean
09:02	between the first term and the fifth term . So
09:08	we should get 12 as an answer . So the
09:13	average of one in 51 plus five or six divided
09:16	by two . A street . So we should get
09:18	a sub three . The first term is three .
09:22	The last term or the fifth term is 48 .
09:26	Now , What's three times 48 ? If you're not
09:29	sure what you could do is break it up into
09:32	smaller numbers . 48 is three times 16 Three times
09:37	three is 9 . So you have the square root
09:39	of nine times the square root of 16 . The
09:42	square root of nine . History . The square root
09:45	of 16 is 43 times four is 12 . So
09:50	the geometric mean of three and 48 Is the middle
09:55	number in the geometric sequence , which is 12 .
10:04	Now , sometimes you need to be able to write
10:06	equations between terms within a geometric sequence . For instance
10:13	, if you want to relate the second equation two
10:16	, the first equation , you need to multiply by
10:17	our I mean the second term to the first term
10:22	. If you want to relate the fifth term to
10:25	the second term , you need to multiply by our
10:27	cube To go from the second term to the 5th
10:30	term . You need to multiply it by our three
10:33	times . If you multiply six by , are you
10:36	gonna get 12 ? If you multiply 12 by are
10:39	you get 24 24 by our You get 48 .
10:43	So to go from the second term to the fifth
10:45	term you need to multiply by our cube . And
10:49	the reason why it's Cuba is because the difference between
10:51	five and to a street and you can check that
10:59	. So if you take the second term which is
11:01	six multiplied by two to the third . That's six
11:04	times eight which is 48 . And that gives you
11:06	the 5th time . So if I want to relate
11:13	the ninth term To the 4th term , how many
11:16	are values do I got to multiply the fourth time
11:18	to get to the nighttime ? 9 -4 is five
11:23	. So I got to multiply the fourth term by
11:25	our to the fifth power to get the ninth term
11:29	. So make sure you know how to write those
11:30	formulas . So we've discussed calculated to some of a
11:40	finite series . Just review if you want to calculate
11:44	the sum Of a finite series , one that has
11:47	a beginning and an end . You would use this
11:51	formula now . What about the sun of an infinite
11:55	? Serious ? How can we find that ? What's
12:00	the formula that we need to calculate us to infinity
12:05	? It's basically this same formula . But without that
12:08	part it's a 1/1 minus R . So here's two
12:18	examples of an infinite geometric series . This is one
12:21	of them and this one is going to be another
12:24	one , 84 21 one half and so forth .
12:33	We can't calculate the sum of both infinite geometric series
12:38	For this one . RS 2 . So our or
12:43	rather the absolute value of R . Is greater than
12:45	one . When that happens , the geometric series diverges
12:52	which means you can't calculate the sum because it doesn't
12:58	it doesn't converge to a specific value . If you
13:01	keep adding these numbers it's not gonna converged to a
13:05	value . It's going to get bigger and bigger and
13:07	bigger . So the series diverges . If you try
13:12	to calculate it let's say you plugged in one for
13:14	a one and mhm to for our it's not going
13:19	to work , you get three over negative one which
13:22	is negative three . And clearly that's not the some
13:24	of this series . The fact that you get a
13:29	negative some from positive numbers tells you something is wrong
13:32	. So this formula doesn't work if the series diverges
13:37	, it only works if the series converges and that
13:39	happens when the absolute value of our is less than
13:42	one . If we focus on this particular infinite geometric
13:47	series , notice the value of our if we take
13:51	the second term divided by the first term , 4/8
13:55	is one half . If we take the third term
13:57	divided by the second term To over four reduces 2
14:01	1/2 . That's the value of our . So for
14:06	that particular series we could say that the absolute value
14:10	of art , which is one half , that's less
14:12	than one . Therefore the series converges , which means
14:19	we can calculate a sum . It has a finite
14:22	some . Even though the numbers get smaller and smaller
14:24	and smaller , now let's calculate the sun . So
14:31	the sum of an infinite number of terms of this
14:35	geometric series is going to be the first term .
14:37	A suborn which is a Over 1 -2 . Where
14:41	are is a half ? 1 -1 half is one
14:46	half . So multiplying the top and bottom by two
14:49	, we get 16 on top . These two will
14:52	cancel . We get one . So the some of
14:55	this infinite geometric series that converges is 16 . So
15:02	that's how you can calculate the sum of an infinite
15:04	geometric series . The series must converge and for that
15:09	to happen , the absolute value of art has to
15:11	be less than one . If it's greater than one
15:14	, the series will diverge and you won't be able
15:16	to calculate the sum . Now let's work on some
15:19	practice problems . Right ? The first five terms of
15:23	each geometric sequence shown below . So let's start with
15:28	the first one . The first term is to to
15:33	find the next term . We need to multiply the
15:37	first term by the common ratio . The second term
15:39	is equal to the first term times the common ratio
15:43	. So two times 3 is six . And then
15:46	to get the third term , we just got to
15:48	multiply the second term by the common ratio , six
15:51	times 3 is 18 , 18 . Times Street is
15:54	54 And then 54 times street . That's 1 62
16:00	. So that's the answer for number one . Now
16:07	, let's move on to number two . The first
16:12	term is 80 . The common ratio is 1/2 .
16:16	So we're gonna multiply 80 by a half , Half
16:19	of 80 is 40 , Half of 40 is 20
16:22	, Half of 2010 . Half of tennis five .
16:25	So those are the first five terms for the second
16:28	geometric sequence . Now let's move on to number three
16:34	. So the first term is six to find the
16:36	next term , we need to multiply six by -2
16:40	. So this is gonna be negative 12 negative 12
16:43	times negative two is positive 24 . And then it's
16:46	just going to alternate . So whenever you see a
16:50	sequence , a geometric sequence with alternative science , then
16:55	you know that the common ratio must be negative number
16:59	two . Right . The first five terms of the
17:02	geometric sequence defined by the recursive formula shown below .
17:08	So we're given the first term When N is too
17:12	we have that . The second term is equal to
17:15	negative four time is the first term , and we
17:19	know that the second term is the first term .
17:21	Times are . So therefore are the common ratio must
17:26	be negative for so any time you need to write
17:30	a recursive formula of a geometric sequence , it's going
17:33	to be a sub N is equal to our Times
17:37	. The previous term , a 7 -1 . The
17:40	next term is always the previous term times the common
17:42	ratio . Mhm . So the common ratio is this
17:50	number -4 . So once we have the first term
17:56	in the common ratio , we can easily right out
17:57	the sequence . So the first term is negative three
18:00	. The second term will be negative three times negative
18:03	four , which is 12 . The third term will
18:07	be 12 times -4 , which is -48 . The
18:13	fourth term is negative 48 times negative four , Which
18:17	is 192 . And then the 5th term 92 times
18:23	negative four . His negative 7 68 . So that's
18:29	how we can write the first five terms of the
18:30	geometric sequence defined by recursive formula . It's by realizing
18:34	that this number is the common ratio , write a
18:39	general formula that gives the f term of each geometric
18:43	sequence And then calculate the value of the 8th term
18:47	of each of those geometric sequences . So let's start
18:52	with number one . So we have the number 6
18:58	24 , 96 3 84 and so forth . The
19:04	first thing we need to do is calculate the common
19:06	ratio . So let's divide the second term by the
19:10	first term , Dividing 24 x six . We get
19:16	four . Now just to confirm that this is indeed
19:23	a geometric sequence , let's take the third term and
19:27	divided by the second term , not the first one
19:31	, So 96 divided by 24 And that is also
19:37	equal to four . So we have a geometric sequence
19:40	here in order to write the formula . All we
19:47	need is the value of the first term and the
19:49	common ratio . So we could use this equation .
19:53	The f term is going to be equal to the
19:55	first term times are Race to the N -1 .
20:00	The first term being six . R . is four
20:06	, So we can write it as a . seven
20:08	is equal to six Times for Race 2 & -1
20:13	. So this is the answer for part A .
20:15	For the first sequence . Now let's move on the
20:22	part B . Let's calculate the value Of the 8th
20:25	term . So we just gotta plug in eight into
20:28	end . So it's six times 4 . Race to
20:32	the 8 -1 . 8 -1 is seven . four
20:37	. race to the seven power is 16,384 Times six
20:44	. This gives us 98,304 . So that is the
20:53	value of the 8th term and you could confirm it
20:58	. If you keep multiplying these numbers by four ,
21:03	you're going to get it 3 84 Times four ,
21:07	that's 1536 . That's the fifth term . A few
21:12	times about four . again You get 61 44 Times
21:18	four You get 24 576 And then times four gives
21:24	you this number . Now let's move on to number
21:29	two . So we have the sequence five -1545 -135
21:46	and so forth . So the first term is five
21:51	. The common ratio which can be calculated by taking
21:56	the second term divided by the first term . That's
21:58	-15 divided by five . That's negative three . Are
22:03	is also equal to the third term divided by the
22:05	second term . So that's 45 Over -15 which is
22:11	-3 . So the value the first term is five
22:19	are is negative three . So now let's go ahead
22:25	and write a general formula that gives us the end
22:28	after . So a seven is going to be a
22:32	someone times are raised to the N -1 . The
22:36	first term is five . R is -3 . So
22:42	this right here is the answer for part A Now
22:46	part B calculate the value of the 8th term .
22:49	So let's replace and with eight negative three raised to
23:01	the center of power . That's -2187 . Multiplying that
23:08	by five . This gives us negative 10,000 935 .
23:15	So that's the final answer . For part B .
23:18	Number four describe each pattern of numbers as arithmetic or
23:22	geometric finite or infinite sequence . Or serious . Just
23:28	looking at the 1st 1 . Do we have a
23:30	common difference or common ratio ? Going from 4 to
23:34	8 ? We increased by four . From 8 to
23:37	12 . That's an increase by four And 12 -
23:40	16 . So we're constantly adding for we're not multiplying
23:44	by four . So therefore we have a common difference
23:49	and not a common ratio . So because we have
23:52	a common difference , this is arithmetic , not geometric
23:59	. We're dealing with addition rather than multiplication . Now
24:04	is this a sequence or series ? We're not adding
24:07	numbers . So we have a sequence . If you
24:13	see a comma between the numbers , it's gonna be
24:15	a sequence . If you see a plus you're dealing
24:18	with a series . Now , is this sequence finite
24:22	or infinite . It has a beginning and it has
24:25	an end . We don't have dots to indicate that
24:28	it goes on forever . So this is finite .
24:31	So we have a finite arithmetic sequence for number one
24:36	. Now let's move on to number two , going
24:39	from 90 to 30 . That's a difference of negative
24:42	60 Going from 30 to 10 . That's a difference
24:45	of -20 . So we don't have a common difference
24:47	here . If we divide the second term by the
24:51	first term , This reduces to 1/3 . If we
24:54	divide the third term by the second term , that's
24:57	also 1/3 . So what we have here is a
25:01	common ratio rather than a common difference . So the
25:06	pattern of numbers is geometric , not arithmetic . We're
25:10	multiplying by 1/3 to get the second term from the
25:13	first term . Now , are we dealing with a
25:17	sequence or serious ? So we don't have a plus
25:21	sign between the numbers . So we have a comma
25:23	so dealing with a sequence and this sequence has no
25:28	end . It goes on forever . So what we
25:31	have here is an infinite geometric sequence . Now .
25:37	For number three , we could see that we have
25:43	a common ratio of two . Five times two is
25:47	10 , 10 times two is 2020 times two is
25:50	40 . So this is geometric . Now there's a
25:57	plus between the numbers . So this is going to
25:59	be a series , not a sequence . And this
26:03	sequence , I mean the series rather comes to an
26:05	end . The last number is 80 . So it's
26:07	not infinite , but it's finite . So we have
26:10	a finite geometric series For the last one . We
26:15	can see that we have a common difference of -450
26:19	-4 . It's 46 46 runners four is 42 .
26:24	So this is going to be arithmetic . We have
26:29	a plus between the numbers so it's a series and
26:33	it goes on forever . So it's infinite . So
26:38	we have an infinite of reference six series number five
26:43	Find the sum of the 1st 10 terms of the
26:45	geometric sequence shown below . So the first term is
26:52	seven . The common ratio -14 divided by seven .
26:58	That's negative too . 28 divided by -14 is also
27:03	negative too . So now that we know the first
27:06	time in the common ratio , we can calculate the
27:08	Sun using this format . So it's the first term
27:12	times 1 - Our race to the end Over 1
27:16	-2 . To the some of the 1st 10 terms
27:20	. It's going to be the first term seven times
27:22	one minus negative two . Now it's a race to
27:27	the end power and is 10 And then divided by
27:31	1 -2 . Negative two . Raised to the 10th
27:44	power . That's positive 1,024 . And here we have
27:51	one minus negative two which is one plus two .
27:54	And that stream 1 -1024 . That's negative . 1000
28:02	23 . Seven times negative 10 23 . Divided by
28:10	three . That's -2387 . So that's the final answer
28:20	. Travis problem . Find the some of the infinite
28:22	geometric series . So we have the numbers 270 ,
28:29	90 30 10 and so forth . So we can
28:36	see that the first term A someone That's 270 .
28:41	The common ratio . If we divide 90 x 70
28:45	, what does that simplify too ? I mean 90
28:48	x 2 70 . Well , we could cancel a
28:51	zero . So we get 9/27 . 9th Street Time
28:55	Street 27 is nine Times Street . That becomes true
28:59	over nine three . We can write that is three
29:03	times 19 is three Times street , this is one
29:05	third . So that's going to be the common ratio
29:11	. And of course if you divide 30 x 90
29:14	, you also get one third . So now that
29:18	we have the first term and the common ratio ,
29:21	we can now calculate the some of the infinite geometric
29:27	series . Using this formula , it's going to be
29:30	a sub 1/1 -2 . By the way , this
29:36	particular infinite geometric series , there's a converge or diverge
29:41	And the absolute value of our is less than one
29:44	. It's 1/3 , which is about .333 or .3
29:48	repeating . So because it's less than one , the
29:52	infinite geometric series converges . We can calculate the sum
29:58	the sum is finite . So let's go ahead and
30:02	calculate that some . The first term is 2 70
30:06	are is one third . So what's 1 -1 3
30:12	1 ? If you multiply it by three of the
30:14	three , You get through every 3 -1 of the
30:18	three which is two or three . Now what I'm
30:24	gonna do is I'm gonna multiply the top and bottom
30:26	by three . These stories will cancel . So it's
30:32	gonna be 2 70 Time Street divided by two to
30:42	70 divided by two . Is 1 35 1 35
30:45	Times Street . That's gonna be 405 . So that
30:51	is the sum of this infinite geometric series .

Summarizer

DESCRIPTION:

This video provides a basic introduction into arithmetic sequences and series. It explains how to find the nth term of a sequence as well as how to find the sum of an arithmetic sequence. It also discusses how to distinguish a finite sequence from an infinite series. It also includes a few word problems.

OVERVIEW:

Geometric Series and Geometric Sequences - Basic Introduction is a free educational video by The Organic Chemistry Tutor.

This page not only allows students and teachers view Geometric Series and Geometric Sequences - Basic Introduction videos but also find engaging Sample Questions, Apps, Pins, Worksheets, Books related to the following topics.