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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