01 - Intro to Sequences (Arithmetic Sequence & Geometric Sequence) - Part 1 - Free Educational videos for Students in K-12

01 - Intro to Sequences (Arithmetic Sequence & Geometric Sequence) - Part 1 - By Math and Science

Transcript


00:00	Hello . Welcome back . We're embarking on an entirely
00:03	new topic from what we have learned recently . We're
00:06	embarking on the topic here in the beginning of sequences
00:09	in math . And then we will be transitioning into
00:11	the related topic of series . So what we're gonna
00:14	do in the beginning is make sure you understand what
00:16	the sequences uh to solve problems involving sequences will transition
00:20	into series . We'll talk about the sigma notation with
00:23	the big , the big capital e greek letter E
00:25	that you see in math . And learn about the
00:28	applications of those ideas . Now in the biggest biggest
00:31	picture people want to know in the very beginning is
00:33	why do I care about sequences in series and math
00:36	? It's really hard for me to tell you exactly
00:38	why because it involves future math things that you haven't
00:41	learned yet . But I'll tell you in a nutshell
00:43	, the foundations of calculus , which all modern science
00:47	engineering and math calculations are done using the concept that
00:50	we learned later in calculus . The foundations of calculus
00:53	really are built on the concept of sequences in series
00:56	. Because later on in calculus we will learn how
00:58	to look at a curve and figure out how fast
01:01	that curve is changing . In order to do that
01:03	, we'll have to zoom in on that curve and
01:04	look at what's happening at a small region . We
01:06	will also take a function , a curve and we'll
01:09	be calculating the area underneath the curve to kind of
01:12	add up what is really going on to be able
01:15	to calculate things like trajectories in physics and other related
01:19	concepts . So the applications of calculus are really limitless
01:22	. In order to do the main ideas of calculus
01:24	, we have to understand what the sequences and we
01:27	have to understand what the series is because that those
01:29	concepts and calculus are built upon . These . The
01:32	good news is sequences and series are very simple to
01:35	understand a sequence which is the topic In this lesson
01:39	we're gonna talk about introduction to sequences and specifically ,
01:42	we'll be talking about the introduction to the arithmetic and
01:45	the geometric sequence . Alright . A sequence is just
01:49	a listing of numbers . That is it . Right
01:51	. If there is no pattern to the listing of
01:53	numbers then the sequence has no pattern . But there
01:56	are some patterns that exist and pop up in nature
01:58	called the arithmetic sequence and the geometric sequence . We're
02:02	going to study those in detail in future lessons .
02:04	This is just an introduction to the general concept .
02:07	So in general a sequence which is where this entire
02:12	journey begins is just a list of numbers in your
02:18	math book . You'll see a much more longer definition
02:20	. But ultimately , when you boil it down ,
02:22	it's just a list of numbers . So , when
02:24	would we care about sequences ? What is a practical
02:27	example of the sequence ? Let's talk about the temperature
02:31	in this room . The temperature in this room every
02:33	day at five p.m. I come in every day at
02:35	five o'clock in the afternoon or in the evening and
02:38	I measure the temperature and I write it down the
02:40	next day . I do the same thing again and
02:41	again . So if I'm going to take a look
02:43	at a practical example , I might look at the
02:45	temperature at five p.m. In this room and the temperature
02:50	might go something like this on day one , it's
02:53	15 degrees Celsius and then 17 Celsius and then 14
02:58	Celsius and then 13 uh Celsius , let's do 15
03:06	Celsius like this , Right ? And so I might
03:09	list these numbers on day one , day , two
03:11	, day three , day four , day five ,
03:13	and then I might say , you know , uh
03:15	dot dot dot and then I might have some final
03:18	temperature , T seven . Okay . So I guess
03:21	I should have put this probably on the same line
03:23	. You have a list of numbers . Maybe you
03:24	get more and more and more numbers and then you
03:26	have T seven . The T seven is the temperature
03:29	at some day in the future . Usually in sequences
03:32	we use the letter N . To tell us on
03:36	whatever day we're doing the measurement in this case ,
03:38	or from taking some measurements every every second . It's
03:42	the basically in tells you what number the measurement is
03:45	that you that you have . And just means it's
03:47	the index that tells us what number of measurement we
03:50	have . So this would be day number 123456 and
03:53	so on and so forth . And so uh this
03:57	first guy uh is all of these guys are what
04:00	we call terms of the sequence . Right . So
04:03	all of these things are what we call terms of
04:08	the sequence ? You're gonna see me Right CQ for
04:11	sequence quite a bit . All right . So for
04:14	this first guy , this first one right here ,
04:18	this is T sub one . It's the very first
04:22	term in the sequence . Right . And you might
04:25	guess that they're on next door is T sub two
04:28	, and this one is T sub three and this
04:31	one is T sub four . You can see the
04:34	progression dot dot dot and then I'm gonna have some
04:37	temperature measurement . So many days down the road .
04:40	T . So , what it means is the sequence
04:43	might have only three terms . In which case it
04:45	has just these first three numbers . These and these
04:48	guys don't exist . Right . If I go out
04:50	to five days , then the term I might have
04:52	five days in my sequence . Okay ? And if
04:56	I go at seven days I'll have some more terms
04:58	in the sequence . And so I generalize it and
05:00	I say the 10th term is just the last term
05:03	at the very end of the sequence . That's all
05:04	it means T seven . Right now , if you
05:07	look at this thing and try to figure out what
05:09	pattern you have , you'll realize quickly that there is
05:12	no pattern to the to these numbers . So I'm
05:14	gonna put that right here . There's no pattern ,
05:19	there's no pattern here . So 15 plus two is
05:21	17 . Okay ? But then 17 you're going down
05:25	to 14 and you're going down by 17 minus three
05:29	is 14 . So you're going down by a different
05:30	amount , then you go down by one , then
05:32	you go up by two . So the numbers are
05:34	jittering around all over the place . And that is
05:36	because the temperature in a room is , I don't
05:39	wanna say it's random , it depends on the weather
05:41	or in the air conditioner , but it's not something
05:43	that's gonna follow a logical progressive pattern , like ,
05:46	like this . Okay , so we would say that
05:48	there's no pattern in this sequence sequences like this are
05:51	very important in the real world . There's lots of
05:52	processes like noise uh like the legitimacy of the stock
05:57	market that aren't really , you can't really predict what's
06:00	going to happen in the future , but there are
06:02	a lot of patterns that do follow a regular pattern
06:04	. Okay , let's take a look at another uh
06:07	type of sequence . Uh Just to get an idea
06:10	, let's take a look at the value of some
06:13	investment that I have . What I mean by that
06:18	is I might invest some money into a stock ,
06:21	let's say , I'm gonna buy some google right and
06:24	I'm gonna put $100 into google . Well that price
06:27	of that stock is going to go up and down
06:29	each day and it's kind of not predictable what's gonna
06:32	happen . It depends on the stock market , depends
06:34	on the economy , it depends on on the the
06:38	amount of fear and greed in the stock market on
06:41	that particular day . So it's gonna kind of go
06:43	bouncing up and down a little bit . So if
06:44	you were to look at the sequence that is because
06:48	of some value of an investment on day one you
06:51	might have it might be worth $100 but on day
06:54	two might be 100 and two . You're happy ,
06:56	you gain some money And you're really happy . It
06:59	might go to one , then it starts going down
07:03	in one of five , You know 101 and then
07:08	you might have some days down the road . Call
07:10	it T . Seven . You need to get used
07:12	to seeing this T . Seven being out there .
07:14	So these are the terms of the sequence and you
07:18	label them just like we did the ones before .
07:21	So this will be term number one , term number
07:23	two , term number three , term number four ,
07:26	term number five . And you might have more measurements
07:29	but eventually you're going to get to some term in
07:31	some however many measurements out in the future . And
07:34	again there's no patterns of this one . So I'll
07:36	put no pattern To figure out a pattern . You're
07:40	looking at the numbers and you're just trying to figure
07:42	out some rule to allow you to predict ahead of
07:45	time what the numbers are going to be . But
07:47	these are random all over the place . I mean
07:49	yeah they're all around $100 . But there are some
07:51	up some down and you can't predict ahead of time
07:53	what it's going to do . If you could predict
07:55	the stock market , you would be rich , nobody
07:57	can do that . So let's talk about something that
08:00	actually does have a pattern . And so that's gonna
08:03	be our first what we're gonna end up calling arithmetic
08:06	sequence . Alright so let's talk about some bank account
08:11	, not an investment , just some bank account .
08:14	Bank balance , Let's look at some bank balance right
08:20	first you take a look at it and you have
08:22	$3, . Then you have six , Then you have
08:27	nine , Then you have 12 , then you have
08:32	15 dot dot dot dot dot . And then someday
08:36	in the future you have some amount of dollars in
08:38	the bank there . Obviously you would for it to
08:40	be a sequence . That is some kind of pattern
08:43	. You would need to be able to predict ahead
08:45	of time what the future balance would be . Now
08:48	this one looks quite a bit different because you can
08:50	see what's happening is I'm depositing $3 every day into
08:53	the bank . So this is day one day two
08:55	day three , day four . And so if I
08:57	wanted to label it I would say this is team
09:00	sub one . This is term number two , term
09:03	number three , term number four , term number five
09:07	and then term in and I just kind of spilled
09:10	the beans . But what's happening here is every day
09:13	I'm adding an additional $3 in this is a predictable
09:16	pattern . I know what the pattern is and because
09:18	that I can go predict . Uh this thing goes
09:21	up to t . Sub five , the fifth measurement
09:23	on my bank account . But knowing the pattern I
09:26	can predict . T . Sub 15 if I want
09:28	to . I can go predict t sub 25 if
09:30	I want to . I can predict T sub 100
09:33	if I want to . I can predict as far
09:34	out in the future as I want . Because I
09:36	know what the pattern is . Uh And just to
09:39	spell it out , what's happening here is you look
09:41	at adjacent terms . We talk about adjacent terms .
09:44	We just mean the terms that are right next to
09:46	each other , right ? So if you look at
09:48	this term and this term what you figure out between
09:51	this term and this term ? The difference between here
09:54	we're gonna call it D . The difference D .
09:55	Is how many dollars ? Six minus three . That's
09:57	$3 . I'm gonna drop the dollar signs from here
10:00	on out to make it easy . What is the
10:02	difference between this term in this term ? 9 -6
10:04	. Again , that's the difference being $3 . What
10:08	is the difference between this ? You see the same
10:10	thing . The difference is $3 . The difference is
10:13	$3 . Because every time I'm subtracting I'm getting exactly
10:16	the same difference . That is what constitutes the pattern
10:20	. This one was very simple because it's just small
10:22	numbers . But ultimately what you , what you want
10:24	to look for is look at all the adjacent terms
10:27	and see if there is a , what we call
10:29	common difference . The difference is $3 and it's common
10:33	to all of the terms . So the sequence is
10:35	governed and defined by that common difference . So this
10:40	kind of sequence that basically is governed by a common
10:44	difference between terms . That is one of the most
10:46	important sequences you'll ever learn about in math . It's
10:49	called the arithmetic sequence . You can think of arithmetic
10:52	meaning like arithmetic , arithmetic is usually addition and subtraction
10:57	. That's what you think of when you think of
10:58	arithmetic . Right ? So the way you remember it
11:01	is arithmetic sequences are just sequences where I'm adding a
11:04	constant number to each term in subsequent terms . So
11:08	that's what in arithmetic sequences . So we need to
11:10	kind of write this down and make sure that we
11:11	are all together . All right . So what we
11:15	know is that this is called in a riff arithmetic
11:21	sequence . Okay . And this is when terms adjacent
11:28	terms . When I say terms , I mean the
11:30	terms right next to each other , they only differ
11:35	by a common difference . Yeah , I'm going to
11:43	underline this because it's the most important concept . The
11:45	common difference . That's the term . That's the idea
11:47	you're going to see in your book . So the
11:49	common difference in this case is D is equal to
11:53	three in this case in this example . So the
11:58	definition of an arithmetic sequence and your book that you'll
12:01	probably see is something like this . It will have
12:03	a lot more words and a bunch of stuff that
12:05	doesn't really matter . But ultimately all it means is
12:08	a listing of numbers where the difference between the numbers
12:12	, the numbers that are next to each other is
12:14	always the same thing . So the difference here was
12:17	three . The difference here was three . The difference
12:18	here was three and the difference here was three .
12:20	So it's arithmetic . If the difference was 333 and
12:24	then right here the difference was four , then this
12:26	would not be an arithmetic sequence anymore because in order
12:29	to be arithmetic , every number in the sequence has
12:33	to differ by a constant number . So if this
12:35	difference by three then three , then this difference by
12:37	one and this difference by three then it's not arithmetic
12:40	anymore . So it's very simple . All you have
12:42	to do is check every pair of numbers and see
12:44	if they all differ by a constant difference . And
12:47	you need to write that constant difference down . That
12:49	common difference down . We call it D . All
12:52	right . And of course the example here being the
12:55	bank account balance of depositing $3 every day . That's
12:58	a relatively easy to understand idea behind geometric sequences .
13:04	I'm sorry , arithmetic sequences . So that was the
13:07	arithmetic sequence . Now we need to kind of touch
13:09	on and introduce the cousin to the arithmetic sequence and
13:13	that's called a geometric sequence . So if we think
13:15	about it when you have an arithmetic sequence it means
13:18	arithmetic . Right . All you're doing is adding a
13:21	common number two to the you're adding a constant common
13:26	difference . You're adding a number to each subsequent term
13:29	. That's all you're doing . Also d here can
13:32	be negative So here we're adding three but it would
13:34	be perfectly fine if the were equal to negative three
13:37	. Negative three negative three negative three . That would
13:39	mean the numbers were going down in the bank account
13:41	bound . So it might be you know , 100
13:44	then 97 . And then you subtract three and subtract
13:46	three and subtract three . That's still arithmetic whether the
13:49	numbers are going up or the numbers coming down .
13:51	The common difference can be positive or negative . That's
13:54	totally fine . It's still arithmetic . Now what do
13:56	you think ? Another kind of sequence might be worth
13:59	studying a sequence that doesn't involve addition or subtraction of
14:02	adjacent terms . It might make sense that we might
14:05	study terms and sequences that don't differ by addition and
14:09	subtraction but that differ by multiplication . And that is
14:13	an incredibly important sequence . Also , it's called the
14:15	geometric sequence . All right , So let me write
14:19	it down and give you a practical example that was
14:23	arithmetic sequence right here . Now we have the geometric
14:29	sequence underline this , and this means that adjacent terms
14:36	means terms that are right next to each other .
14:39	I'm going to reword it a little bit , but
14:41	ultimately what it does mean is that those adjacent terms
14:44	have a common multiplier that goes between the terms .
14:48	So instead of a common difference , it's a common
14:50	multiplier . Keep that in the back of your mind
14:52	. I'm gonna write it down more like your book
14:53	will talk about it and then we'll give an example
14:55	would be crystal clear . Adjacent terms have a common
15:02	You might think I'm gonna write down common multiplier ,
15:04	but I'm gonna write it down as common ratio .
15:06	This is what you're gonna see in your books .
15:07	I'm gonna teach you the way all the books talk
15:09	about it , but I'm gonna show you how easy
15:10	it is to understand . And the common ratio is
15:13	not called D . It's called R because R for
15:16	ratio . Right . So , that means pairs of
15:22	terms have a common multiplier . All right . This
15:34	is what the definition of a geometric sequences . Now
15:37	, let's practically talk about it because it's actually much
15:40	easier to understand than this thing would leave you believe
15:42	. What if I give you a a sequence that
15:46	looks like this two comma six , comma 18 ,
15:51	comma 54 . This is the sequence and it's a
15:55	good idea in your mind , at least to say
15:58	that this is term one term to term three in
16:00	term four . Just like I've listed them here ,
16:02	but I'm not gonna write down the terms underneath .
16:04	You just need to know that that first term is
16:06	term one . The next one is two , then
16:08	three , then four and so on . So is
16:10	this an arithmetic sequence ? Yes or no ? All
16:12	you have to do is say am I adding a
16:14	constant number each time ? Well , the difference between
16:16	here and here is four and the difference between here
16:19	and here is definitely not for and the difference between
16:21	here and here is definitely not for . So it's
16:23	definitely not an arithmetic sequence . The next thing you
16:26	have to ask yourself is is there a common multiplier
16:28	between terms ? Well , if I look at this
16:31	two times three is six , so I'm gonna multiply
16:34	by three and then six times three is also 18
16:38	. So this is also multiplied by three . And
16:40	then if you get your calculator at 18 times three
16:43	is also 54 . So for the terms in the
16:45	sequence from everything that we have written down , this
16:48	is what we call a geometric sequence . Because the
16:51	terms have a common multiplier , it allows me to
16:54	predict the terms in the future . Remember I said
16:57	in order for the sequence to have any usefulness ,
17:00	I have to be able to predict down the road
17:02	and knowing this rule , I can predict again term
17:05	number four , term number five , term number 10
17:07	. If I want to just keep multiplying by three
17:09	to get all the terms in the sequence and I
17:11	can write them all down . All right . So
17:14	if this common multiplier thing is here , then why
17:16	does this definition talk about a common ratio ? Because
17:20	when you think about what is the word ratio ?
17:21	Mean ratio is the division of things . So when
17:24	two terms have a common ratio , it's exactly the
17:28	same thing is saying that they have a common multiplier
17:30	. This common multiplier wording is my wording . It's
17:33	not worrying that you'll see in a book most of
17:35	the time , you'll just see that the geometric sequence
17:38	has a common ratio . So here's what I mean
17:40	by that notice I said times three times three times
17:43	three . But let's look and see what the ratio
17:45	of 6 to 2 is , because that's what a
17:47	ratio is . What is six divided by two .
17:49	Take this divided by this , what what do you
17:50	get ? You get three ? Let's look at this
17:53	common ratio 18 divided by six . What do you
17:55	get ? You get three ? Let's look at this
17:58	common ratio , 54 divided by 18 . What do
18:01	you get ? You get three ? So you see
18:03	the common ratio between these two terms is three .
18:06	The ratio between these two terms is three . The
18:08	ratio between these two terms is three . The common
18:11	ratio are in this case is equal to three .
18:14	That is what you're gonna need to write down when
18:16	you're defining the geometric sequence . You want to figure
18:19	out what the common ratio is . It's exactly the
18:21	same thing as saying what is the common multiplier ,
18:24	but they don't usually phrase it that way in books
18:27	. They want you to tell them what the common
18:28	ratio is . That's okay . That's all you're doing
18:29	when you look at this anyway , we know that
18:31	two times three is six . The reason we knew
18:33	it is because we memorized it . But really what
18:36	you're doing is you're taking six , divide by two
18:38	and you're getting three , you're taking 18 divided by
18:40	six , you're getting three , you're gonna taking this
18:41	divided by this , you're getting three . So whether
18:44	you think of it as a common multiplier or a
18:47	common division , common ratio between them , it's the
18:49	same exact thing , what you want to know in
18:52	this case that common ratio R . Is three .
18:54	And for an arithmetic sequence , it's not multiplication or
18:58	division at all . It's only if the terms differ
19:01	by some common uh difference which can be positive or
19:05	negative because addition and subtraction are basically the same thing
19:09	. Ultimately . All right , that's all you really
19:12	have to worry about . People stress out so much
19:14	about what an arithmetic sequences versus what a geometric sequences
19:17	that is the difference . So , let's solve a
19:19	couple of quick problems to really get a handle on
19:24	this . What I want to do is I'm gonna
19:27	write down the sequence , and I want us to
19:29	figure out together is an arithmetic sequence . Is it
19:31	a geometric sequence ? And also to predict the final
19:34	two terms of the sequence . Pretty simple stuff .
19:36	Nothing above basic addition or multiplication . So what if
19:40	you have 20 17 , 14 , 11 and then
19:46	some number and then some number right here . I
19:49	want to predict what these are . But before I
19:50	do that I want you to tell me is this
19:52	arithmetic or geometric ? And then how do we predict
19:56	what these are ? So you have to say ,
19:57	well , is it arithmetic ? What I need to
19:59	do is look at the common the commonality in the
20:01	terms I'm going to look at this and say what
20:04	is term number two minus term number 1 , 17
20:06	minus 20 comes out to -3 . D . is
20:10	-3 . The difference , what is the difference here
20:14	? 14 minus 17 . The D is again negative
20:16	three . What is the difference here ? 11 minus
20:18	14 . D is negative three . Remember I told
20:21	you D can be negative . All that means is
20:23	the terms are just subtracted going down like this .
20:26	So the common difference , the common difference difference .
20:33	D . In this case is -3 . So because
20:36	of that we know this is arithmetic sequence . That's
20:41	all you need to know right now in order to
20:44	predict this is term 1234 In order to predict term
20:48	number five And term number six , I need to
20:52	use this definition . Right ? So how do I
20:54	do that ? Well what you say is term number
20:56	five is going to be term number four plus the
21:01	common difference but the common differences negative . So it's
21:03	gonna be 11 minus three . You're adding the common
21:06	difference but the common differences negative . So you're really
21:08	subtracting it . So what you figured out is term
21:11	number five is what do you get there ? Eight
21:14	. Okay . And then in order to figure out
21:17	what term number six is , I need to know
21:19	what term number five is , so it's eight but
21:22	then I have to add that common difference to common
21:24	differences , negative . So it's really a subtraction term
21:26	, number six is five . Yeah , so that's
21:30	what you get right there . Eight and five .
21:32	So if I wanted to write the sequence down ,
21:34	I would write 2017 and 14 and 11 , then
21:37	eight , then five . And you would see every
21:40	one of these have the same uh common difference .
21:43	In fact , if you look at these two that
21:44	we just calculated the common difference five minus eight again
21:47	is negative . Three . It's the same common difference
21:50	. That's all you have to do when figuring out
21:52	. If something is an arithmetic sequence , let's take
21:54	a look at one more and it probably won't be
21:56	a big surprise and it's going to end up becoming
21:58	a geometric sequence , Let's say 1 5 25 .
22:04	25 And then something and something else . I want
22:07	to figure out what these two are . And I
22:08	want you to tell me arithmetic or geometric . Well
22:11	, if I look at the common difference , 5
22:13	-1 is four . The common difference here , 25
22:16	-5 is 20 . So the common differences automatically .
22:18	Not the same . But I can realize that if
22:22	I want to look at the common ratio five divided
22:25	by one , that's five . Right . And then
22:29	what's this ? 25 divided by the next door neighbor
22:32	. five , that's also equal to five . And
22:35	then I can go here . 1 25 Divided by
22:38	25 by its neighbor . That's five . So the
22:41	common ratio , which means the ratio between adjacent terms
22:44	is the same thing . It's five . That means
22:46	to go from here to here . It's the same
22:48	as multiplying by five to go from here to here
22:50	. The same thing is multiplying by by three .
22:53	Sorry about that by five . And then you go
22:56	from here to here is multiplying by five . It's
22:58	a common multiplier . Same thing as a common ratio
23:01	. Yeah . So to write down our answer ,
23:03	we figured out the common ratio . We call that
23:09	our is five . And because it has a common
23:12	ratio , we know it's a geometric sequence . So
23:20	we circle that . That's the first thing we want
23:21	to do . Now we want to predict this is
23:23	term one , term to term three , term four
23:26	. We want to know what term five and six
23:27	are . So what is term five ? What is
23:30	it gonna be ? It's gonna be 1 25 multiplied
23:33	by five again . 1 25 and I cannot write
23:36	today 1 25 multiplied by five . So term number
23:40	five Is when you multiply that you get 6:25 ,
23:44	just double checking myself . And then term number six
23:48	is dependent on its next door neighbor , which is
23:50	turn number 5 625 . Again , common multiplier of
23:54	five And what you're gonna get is 31 25 .
24:00	So this is term number six , this is term
24:02	number five . So if I wanted to write the
24:03	whole thing down would be 15 25 1 25 6
24:07	25 31 25 . And again , the common ratio
24:11	holds . If you grab a calculator and divide 3125
24:14	divided by 6 25 . What are you gonna get
24:16	? You're gonna get that common ratio of five because
24:19	all the adjacent terms differ by that common ratio .
24:22	So in this lesson we have introduced the concept of
24:25	sequences before starting this lesson , you may or may
24:29	not have known what the sequences at all . We've
24:31	now talked about the idea . There's lots of sequences
24:34	. Most of them don't have any real mathematical value
24:37	because they're kind of random temperature in the room value
24:40	of some investment being unpredictable like that . Those are
24:42	also sequences , but they don't , they aren't labeled
24:45	with the term arithmetic or geometric and we can't predict
24:48	ahead of time . What they're going to be In
24:50	arithmetic sequence is like an example of a bank account
24:53	where I'm adding money and every day the common difference
24:56	between terms is just a number . That is what
24:58	defines to be an arithmetic sequence . A geometric sequences
25:01	when you look at the next door neighbor terms and
25:03	they differ by a multiplier , two times three being
25:06	66 times three is 18 . We talked about why
25:09	the division of adjacent terms is called the common ratio
25:11	. There . Now we have a lot more to
25:13	do with this . We're just scratching the surface the
25:15	next couple of lessons , we'll do some more problems
25:17	to give you more practice with the introduction of sequences
25:20	. And then we'll be diving in a lot more
25:21	detail into arithmetic and geometric sequences in the coming lessons
25:25	. So make sure you understand this , then follow
25:27	me on to the next lesson and let's continue cranking
25:29	through sequences in series .

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