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

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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