07 - The Geometric Sequence - Definition & Meaning - Part 1 - By Math and Science
Transcript
| 00:00 | Hello . Welcome back . The title of this lesson | |
| 00:02 | is called geometric sequences . This is part one of | |
| 00:05 | several lessons . So the geometric sequence is something that | |
| 00:08 | you learn usually in somewhere in algebra but it's kind | |
| 00:12 | of there and we use it for some some some | |
| 00:14 | concepts there in algebra but really it doesn't come into | |
| 00:17 | its own is something really useful to know about really | |
| 00:20 | . Until a little bit later in math and pre | |
| 00:21 | calculus and calculus . So when you study this for | |
| 00:24 | the first time you might wonder why do we care | |
| 00:26 | so much about it ? But the reality is is | |
| 00:28 | that we just need to learn it now so that | |
| 00:29 | we can have it in our tool bag for later | |
| 00:31 | math classes . In which case it becomes extremely useful | |
| 00:34 | . Now the good news is geometric sequence is very | |
| 00:36 | simple to understand if you remember back to the arithmetic | |
| 00:40 | sequence . Those were sequences of numbers where basically we're | |
| 00:43 | just adding or subtracting a constant number to get each | |
| 00:46 | of the terms . Like we might add three to | |
| 00:48 | get the next term , then add three , then | |
| 00:50 | add three , then add three . Okay , so | |
| 00:52 | address subtract numbers to get those arithmetic sequences for a | |
| 00:55 | geometric sequence . Basically what we're gonna do is we're | |
| 00:57 | gonna start with term number one and we're gonna start | |
| 00:59 | multiplying instead of adding or subtracting , we're gonna multiply | |
| 01:02 | by a number , multiplied by a number , multiply | |
| 01:05 | it by a number . So the term geometric sequence | |
| 01:07 | just means a sequence of numbers uh where you end | |
| 01:10 | up multiplying to get each subsequent term . Now , | |
| 01:13 | what I'm gonna do first is write down the formula | |
| 01:16 | for all of the terms of the geometric sequence . | |
| 01:18 | You probably won't understand exactly why it's the way that | |
| 01:21 | it is in the beginning . But by the end | |
| 01:22 | of this lesson you will understand exactly why that equation | |
| 01:25 | for the geometric sequence works . And we're gonna solve | |
| 01:27 | several problems in the concept of geometric sequence . So | |
| 01:31 | here is the formula for geometric sequence . Okay , | |
| 01:39 | here , it is not too complicated . The 10th | |
| 01:43 | term down the road is equal to the first term | |
| 01:46 | in the sequence . T sub one multiplied by our | |
| 01:50 | I'll explain what that is in the second raise to | |
| 01:52 | the power of N -1 . So this is an | |
| 01:55 | equation that you will definitely see in , you know | |
| 01:58 | , all algebra books at some point . Pre calculus | |
| 02:00 | calculus and so on . And as I said , | |
| 02:02 | it becomes extremely important , especially in calculus to not | |
| 02:05 | even in calculus one calculus to it becomes extremely important | |
| 02:08 | to understand what the geometric sequences . So the question | |
| 02:12 | that we have is , okay , I can tell | |
| 02:13 | you that this is the equation of a geometric sequence | |
| 02:15 | , but why is the term equal to all of | |
| 02:18 | these things ? So , in order to do that | |
| 02:20 | , we have to write a few things down . | |
| 02:21 | None of them are very hard to understand . All | |
| 02:23 | right . So the first thing is I already told | |
| 02:25 | you the terms of the geometric sequence are basically when | |
| 02:28 | you multiply by a number to continue getting the terms | |
| 02:31 | in the sequence . Another way of saying that the | |
| 02:33 | way that you're going to see it in your textbook | |
| 02:35 | is that the terms of the geometric sequence , they | |
| 02:38 | differ by what we call a common ratio . So | |
| 02:41 | what I actually like to tell you is the terms | |
| 02:43 | of the sequence have a common multiplier , Right . | |
| 02:45 | But you don't see that written in books ? You | |
| 02:47 | see it written as a common ratio . I personally | |
| 02:50 | find it easier to understand thinking about it as a | |
| 02:52 | common multiplier . But I'm gonna show you both ways | |
| 02:54 | to think about it . Both of them are extremely | |
| 02:56 | easy to understand . So what I need to write | |
| 02:58 | down uh first is that the geometric terms are related | |
| 03:08 | by a common racial and almost all books that I've | |
| 03:16 | ever seen . This comin ratio is just called are | |
| 03:19 | in fact , that's why I have are in the | |
| 03:20 | equation . So this are is what we call the | |
| 03:22 | common ratio between the terms . But what I like | |
| 03:24 | to tell people is that it's all related to what | |
| 03:26 | we call a common multiplier . I don't know why | |
| 03:33 | . But for me , it's just easier to think | |
| 03:35 | of things being multiplied to get the future terms rather | |
| 03:38 | than some division , which is what the ratio is | |
| 03:40 | . The ratio is division . All right . So | |
| 03:41 | let's uh understand this by taking a look at a | |
| 03:44 | very simple geometric series . So , this guy is | |
| 03:47 | a geometric series five , then we have 10 . | |
| 03:50 | Then we have 20 and then we have 40 . | |
| 03:53 | And then some dot dot dot down the road . | |
| 03:55 | We have t number 10 . Okay , so we | |
| 03:58 | have term one term to term three , term for | |
| 04:01 | a bunch of terms in the middle . And then | |
| 04:02 | here's the 10th term way down at the end . | |
| 04:04 | First of all . How do you know that ? | |
| 04:05 | This is a geometric series and not an arithmetic . | |
| 04:08 | Uh Sorry about that . Sometimes I say series because | |
| 04:11 | we're gonna be talking about geometric series and and and | |
| 04:14 | such very soon , which are also very , very | |
| 04:16 | important . So , in this lesson , if I | |
| 04:18 | say series , please forgive me and just understand that | |
| 04:20 | . I'm trying to tell you about sequences here . | |
| 04:23 | So , anyway , how do we know this listing | |
| 04:25 | of numbers is a geometric sequence rather than an arithmetic | |
| 04:29 | sequence . When you just look at the terms here | |
| 04:31 | , I've added five to get 10 . But here | |
| 04:34 | we go from here to here , I've added 10 | |
| 04:36 | . But to go from here and here , I've | |
| 04:37 | added 20 . So all of the common differences between | |
| 04:40 | the terms , they're all different . So this is | |
| 04:41 | not an arithmetic sequence . So how do we know | |
| 04:44 | it's a geometric uh how do we know it's a | |
| 04:47 | geometric sequence ? We'll fight to go from here to | |
| 04:49 | here . What do I do ? Well , I'm | |
| 04:50 | going to multiply by 25 times two is 10 . | |
| 04:54 | How do I go from here from here ? 10 | |
| 04:56 | times two is 20 . How do I go from | |
| 04:59 | here to here ? 20 times two . What's not | |
| 05:03 | times 20 times two is 40 . And so it | |
| 05:05 | goes to get the next term after this . I | |
| 05:07 | would multiply by two to get the next term after | |
| 05:09 | that and multiply by two . And I keep multiplying | |
| 05:11 | and multiplying and multiplying eventually I get to term number | |
| 05:14 | 10 or if I want to take it out farther | |
| 05:16 | , I go to term number 100 term number two | |
| 05:18 | , 5000 . You can just keep multiplying by two | |
| 05:21 | by two by two in this case , this is | |
| 05:23 | what I call the common multiplier is a two , | |
| 05:26 | right ? So in this case we say common multiplier | |
| 05:35 | Is equal to two for this particular sequence . Now | |
| 05:39 | let's rewrite the sequence and let's talk about it more | |
| 05:41 | like what you would see in your textbook . This | |
| 05:43 | is the way I like to think about it , | |
| 05:45 | but let's write the same sequence down 5 , 10 | |
| 05:48 | , 20 40 dot dot dot and the 10th term | |
| 05:53 | . Somewhere down the road . All right . Um | |
| 05:56 | what do we mean when we say geometric terms are | |
| 05:59 | related by a common ratio , which is also called | |
| 06:01 | a common multiplier . What is a ratio ? A | |
| 06:04 | ratio is just a division . A ratio of 1 | |
| 06:08 | to 2 means one divided by two , means a | |
| 06:10 | half . A ratio of 2 to 4 means two | |
| 06:14 | divided by four , which we can simplify the one | |
| 06:16 | half . So when you divide two numbers , it | |
| 06:18 | just means you're taking the ratio . So if you | |
| 06:20 | take and look at these first two terms and you | |
| 06:23 | take the second term divided by the first term , | |
| 06:25 | 10/5 , of course you get to . So what | |
| 06:28 | this means is the ratio between these terms is to | |
| 06:31 | the ratio between the first two terms is equal to | |
| 06:33 | two . R is equal to two . Now if | |
| 06:36 | you examine the second two terms , the ratio here | |
| 06:38 | is 20/10 . I'm taking a look only at these | |
| 06:41 | two terms . And so the ratio again is to | |
| 06:44 | same ratio . This ratio right here is again 40/20 | |
| 06:48 | . So the ratio is too and you can see | |
| 06:51 | that going down the line . The ratio between all | |
| 06:53 | of these storms are going to be too . So | |
| 06:54 | it's exactly the same thing . What we say is | |
| 06:57 | that we say the common ratio is equal to uh | |
| 07:05 | to and it's the same common ratio . Alright . | |
| 07:08 | And because of that , what we basically mean is | |
| 07:11 | that to get the next term , you can think | |
| 07:14 | of it as all of the adjacent terms have the | |
| 07:16 | same ratio . When you when you predict the next | |
| 07:19 | term in the future , you need to pick that | |
| 07:21 | term so that when you divide it by the previous | |
| 07:23 | term , you still get to , that's what basically | |
| 07:25 | you do . But to me that just turns my | |
| 07:27 | brain and noodles . Okay , I don't I don't | |
| 07:29 | think of things that way . So for me it's | |
| 07:31 | much easier to take this common ratio and think of | |
| 07:33 | it as a common multiplier . Keep multiplying multiplying multiplying | |
| 07:36 | multiply whatever you get for your common ratio , it's | |
| 07:39 | the same thing as a common multiplier . That's how | |
| 07:41 | you get your future terms . Now I want to | |
| 07:44 | rewrite the sequence one more time over here because I'm | |
| 07:46 | going to draw a very important conclusion five 10 20 | |
| 07:52 | 40 10th one down here . Okay . And we | |
| 07:57 | already said that the common ratio here was to the | |
| 08:02 | common ratio here was to the common ratio here was | |
| 08:05 | to presumably there's some common ratio here and then there's | |
| 08:09 | some common ratio here . All of them are equal | |
| 08:11 | to two between all of these adjacent terms in there | |
| 08:13 | . Okay . Now the question I have for you | |
| 08:15 | is how many ratios do we actually have here ? | |
| 08:18 | Well , if you were to if you have 10 | |
| 08:20 | terms right , that's what I have 10 terms here | |
| 08:22 | then I think you can convince yourself that there's nine | |
| 08:25 | ratios And there is a total of 10 terms . | |
| 08:30 | So do you see how , when you take a | |
| 08:33 | look and see how many ratios you have ? There's | |
| 08:35 | always one less ratio than there is number of terms | |
| 08:39 | . There's 10 terms of course all the way to | |
| 08:40 | number 10 . But because I'm looking in pairs like | |
| 08:43 | this , this is one ratio 2345 And if you | |
| 08:45 | go all the way out to the 10th term there's | |
| 08:47 | always gonna be one less ratio because you're looking at | |
| 08:50 | pairs of terms . So in this case there's nine | |
| 08:52 | ratio . But 10 terms . So how would I | |
| 08:56 | decide or calculate ahead of time ? What would term | |
| 08:59 | number 10 be If I want to calculate what this | |
| 09:02 | is ? Well , if I want to figure out | |
| 09:04 | what it is , I just start with the first | |
| 09:05 | term and multiply by two . That gives me the | |
| 09:08 | second term . Right ? So I could say five | |
| 09:11 | times two . That's gonna give me that second term | |
| 09:13 | . But then to get the next term I just | |
| 09:15 | have to multiply by two again . C5 times two | |
| 09:18 | is 10 times two is 20 . But then I | |
| 09:20 | multiplied by two again and I get to 40 . | |
| 09:22 | You see what's happening here is 123 45678 and nine | |
| 09:30 | . So here's five times two is 10 then 20 | |
| 09:33 | then 40 then 80 . And go all the way | |
| 09:35 | down here . You should have 123456789 multiplication is by | |
| 09:40 | two . Because there are nine ratios . There are | |
| 09:42 | nine multiplication by two . To get over here . | |
| 09:45 | So there are nine of these . So what does | |
| 09:51 | all this mean ? If I want to calculate what | |
| 09:53 | the 10th term actually is , it's gonna be we'll | |
| 09:57 | do an equal sign . It would be five times | |
| 09:59 | two to the power of nine because an exponent means | |
| 10:02 | you're multiplying by itself uh with a total of nine | |
| 10:05 | of America to to the 95 times two to the | |
| 10:07 | ninth power . Now , if you were to run | |
| 10:09 | this through a calculator , the 10th term actually turns | |
| 10:11 | out to be 2560 . That's what it comes out | |
| 10:16 | to be . When you keep multiplying by two over | |
| 10:17 | and over again . This is what you get by | |
| 10:19 | the 10th term . But when I want to turn | |
| 10:21 | your attention back to this , this is the secret | |
| 10:23 | sauce right here . Notice that the 10th term down | |
| 10:26 | in the future was just equal to whatever the first | |
| 10:28 | term was multiplied by this common ratio . But raised | |
| 10:33 | to the power Of however many terms I had -1 | |
| 10:36 | because that's how many multiplication I actually had . So | |
| 10:39 | we can generalize this and we can say that the | |
| 10:42 | 10th term down the road , whatever number term we | |
| 10:44 | want to go to . Not the 10th term , | |
| 10:45 | but just whatever random term we want . We just | |
| 10:48 | start by the first term five and we start multiplying | |
| 10:51 | by twos . How many times do we multiply by | |
| 10:54 | two ? We just do it in minus one times | |
| 10:56 | . Because that's how many Common , that's how many | |
| 10:58 | ratios we're going to have is the term we're trying | |
| 11:01 | to get to -1 . Right ? And this is | |
| 11:05 | five . Of course . So let me replace five | |
| 11:08 | with T . Someone will make it general . Okay | |
| 11:11 | , so we'll say that the term of any geometric | |
| 11:14 | sequence is just equal to the first term of the | |
| 11:16 | sequence times . And instead of two , let's replace | |
| 11:20 | that . And we'll say that that is going to | |
| 11:22 | be whatever the common ratio for that sequences . Alright | |
| 11:26 | , so T one times r to the N -1 | |
| 11:29 | power . T one times r to the N -1 | |
| 11:32 | power . This is exactly what the geometric sequences . | |
| 11:34 | This is why the geometric sequences equal to . This | |
| 11:37 | is why the equation is what it is . It's | |
| 11:40 | because to arrive whatever term you want down the road | |
| 11:43 | for any sequence , you just start with the first | |
| 11:44 | one and start multiplying by this common multiplier , Which | |
| 11:47 | is in all books called the common ratio . And | |
| 11:50 | how many times do you do the multiplication ? Just | |
| 11:51 | in -1 time , whatever term trying to get to | |
| 11:54 | -1 . This is why the geometric sequence equation works | |
| 11:58 | the way that it works . So now we have | |
| 12:00 | enough information to start solving some problems and you're gonna | |
| 12:03 | find out that even though a lot of students will | |
| 12:07 | say , well why do I care about this geometric | |
| 12:09 | sequence ? Who cares ? But the reality of it | |
| 12:11 | is it's kind of nice because none of these problems | |
| 12:13 | are gonna be hard . If you're doing 10 pages | |
| 12:15 | of work for these problems , you're working too hard | |
| 12:18 | . Okay . None of them should require that much | |
| 12:20 | work . So what I want to do for these | |
| 12:22 | equate for these sequences is find an equation for the | |
| 12:26 | 10th term . If you are asked to find the | |
| 12:30 | term they want you to generalize it to give , | |
| 12:33 | to give that equation . So what if you have | |
| 12:34 | 26 18 54 dot dot dot , you can have | |
| 12:40 | , you know , T . Seven sitting out here | |
| 12:42 | and I want to give an equation for T . | |
| 12:44 | Seven so that I could put in the value of | |
| 12:46 | N for whatever term I want and just immediately calculate | |
| 12:49 | what the answer is going to be given the geometric | |
| 12:51 | sequence equation . So the first step is to figure | |
| 12:54 | out what the common ratio is . So what you | |
| 12:57 | do is you take a look at these terms and | |
| 12:59 | you say that the common ratio is this term divided | |
| 13:01 | by this one , So the common ratio is three | |
| 13:04 | And you do the same thing 18 , divided by | |
| 13:06 | six is three . You do the same thing here | |
| 13:09 | . 54 , divided by 18 is again three . | |
| 13:11 | The com the ratio is the same in all cases | |
| 13:14 | . So this is a geometric sequence . If one | |
| 13:17 | of these ratios was different than three , like if | |
| 13:19 | the first two were ratio had a two , but | |
| 13:22 | the third one had a ratio of one or four | |
| 13:24 | or six or whatever . Then it's not a geometric | |
| 13:26 | sequence for to be a geometric sequence . Every pair | |
| 13:29 | of terms in the entire sequence has to have this | |
| 13:32 | common multiplier , or you can think of it as | |
| 13:34 | the common ratio . All right . So , we | |
| 13:37 | need to write down this equation for the term . | |
| 13:40 | The term is always equal to the first term times | |
| 13:43 | whatever the common ratio to the power of the -1 | |
| 13:47 | . All right , But in this case the first | |
| 13:49 | tournaments too , The common ratio is 3 to the | |
| 13:52 | power of in -1 . Okay , so you just | |
| 13:56 | write down t uh 72 times three in minus one | |
| 14:01 | . And you have to be really careful here to | |
| 14:02 | make sure you understand this in minus one . Exponent | |
| 14:05 | only applies to the three . You're getting that as | |
| 14:08 | a number and then you're multiplying the result times two | |
| 14:11 | . So , this is going to give you whatever | |
| 14:13 | term you want in the sequence . Let's go ahead | |
| 14:15 | and just spend a second to check it for a | |
| 14:17 | couple of terms . Let's go over here . Let's | |
| 14:19 | see what happens if we put t someone into this | |
| 14:21 | equation find the first term , right ? So it's | |
| 14:24 | we're saying that it's two times three to the N | |
| 14:27 | -1 . In -1 . Being in is equal to | |
| 14:30 | 1 -1 . We're just putting the value of one | |
| 14:32 | into this equation . Well you're going to get two | |
| 14:36 | times three to the zero so t someone this is | |
| 14:40 | one times two is two and that's the first term | |
| 14:42 | of the sequence . We'll do one more Two times | |
| 14:46 | 3 to the 2 -1 . All I did was | |
| 14:48 | put the number two into this equation . What are | |
| 14:50 | you gonna get two times three to the one power | |
| 14:54 | which you know is six . That is the second | |
| 14:56 | term you could go and put in is equal to | |
| 14:58 | three and is equal to four and recover all these | |
| 15:00 | terms . And then you can go beyond for putting | |
| 15:02 | 567 1001 million in there or whatever and you will | |
| 15:06 | get whatever that term is of the sequence . So | |
| 15:09 | you don't have to , even though I introduce all | |
| 15:11 | of this , I introduce all of this and starting | |
| 15:14 | at the first term and then times two times two | |
| 15:16 | times two and time . That's how you can get | |
| 15:17 | the terms of course . But if you want to | |
| 15:19 | calculate the 1594th term , you're gonna be multiplying a | |
| 15:23 | long time . You don't want to do that . | |
| 15:25 | What you want to do in those cases , figure | |
| 15:27 | out what the equation is and just put the term | |
| 15:29 | number in here calculated and that's what you're gonna get | |
| 15:31 | the answer . Yeah . Alright . Next problem . | |
| 15:35 | What if I have three negative 12 , 48 , | |
| 15:43 | dot dot dot . Do not worry . If the | |
| 15:45 | terms alternate in signs like this , we're gonna see | |
| 15:48 | how that works out . All right . So what | |
| 15:51 | do we get right here ? This common ratio here | |
| 15:55 | is gonna be this divided by this negative 12/3 . | |
| 15:58 | So what do you get negative for Now ? All | |
| 16:01 | of these should have exactly the same common ratio . | |
| 16:03 | Otherwise this is not a geometric sequence at all . | |
| 16:05 | So this ratio is 48 over negative 12 . And | |
| 16:09 | you can see this is going to then be negative | |
| 16:12 | four . This common ratio negative 1 92/48 . To | |
| 16:17 | grab a calculator , gonna find . This is also | |
| 16:19 | equal to negative four . So this is a geometric | |
| 16:21 | sequence as well . So how do we write down | |
| 16:25 | what the equation for the interment ? It's the first | |
| 16:27 | term , I'm sorry . The term is equal to | |
| 16:29 | the first term times the common ratio to the power | |
| 16:32 | of N -1 . The first term was three . | |
| 16:37 | The common ratio . You need to wrap it in | |
| 16:39 | parentheses because its a negative sign . The common ratio | |
| 16:41 | is negative four and then you have the N -1 | |
| 16:44 | right here . So you say the in term Is | |
| 16:48 | three times negative four . In -1 . It's really | |
| 16:52 | important when you're comin ratio happens to be negative . | |
| 16:55 | This is the thing that has to go in the | |
| 16:57 | base of the of this expanded here . So you | |
| 16:59 | need to wrap it in parentheses . So the negative | |
| 17:01 | sign is kind of the exponent is also acting on | |
| 17:03 | the negative side as well . Now let's just take | |
| 17:06 | a second to check these or at least check a | |
| 17:08 | couple of them . Let's go in and take a | |
| 17:09 | look at what would happen if we take a look | |
| 17:11 | at t sub one we're saying it's three times negative | |
| 17:15 | four and the power is in minus one . So | |
| 17:19 | we're gonna put 11 minus one . So this is | |
| 17:21 | zero . So what you're gonna get here is zero | |
| 17:24 | negative four to the zero . Power is one times | |
| 17:26 | three is three . That's the first term in the | |
| 17:28 | sequence . The second term is three multiplied by negative | |
| 17:34 | four to the power of in minus one . I'm | |
| 17:37 | putting in is equal to two to minus one . | |
| 17:40 | So what you're gonna have here is three times negative | |
| 17:42 | four because the exponent is just one . So then | |
| 17:45 | you get negative 12 -12 is the second term of | |
| 17:48 | the sequence . So keep on cranking through and you | |
| 17:50 | can recover all of these terms and you can go | |
| 17:52 | beyond it to get to whatever term you want . | |
| 17:56 | Now these are not too bad . Whenever you have | |
| 17:59 | numbers like this that we can easily divide and all | |
| 18:01 | but students sometimes get completely tripped up whenever the sequence | |
| 18:05 | has radicals or other weird things , if you see | |
| 18:08 | radicals in your sequence or if you see fractions in | |
| 18:10 | your sequence or if you see , you know , | |
| 18:12 | who knows what else , you can see X , | |
| 18:14 | some kind of exponents or fractional exponents in your sequence | |
| 18:17 | . The same principles hold in order for these things | |
| 18:19 | to be geometric sequence , you have to divide each | |
| 18:21 | of those terms and make sure that they have the | |
| 18:23 | same common ratio and then you use the exact same | |
| 18:26 | equation . Nothing changes . So don't try to invent | |
| 18:28 | new math . Uh If you see a sequence that | |
| 18:31 | is weird . A good example of that will be | |
| 18:33 | the following . The first time I saw the sequence | |
| 18:35 | , I was like whoa , it's kind of weird | |
| 18:36 | . The first one is one . uh then you | |
| 18:39 | have square root of two , then you have to | |
| 18:42 | then you have two times the square root of two | |
| 18:44 | , then dot dot dot . Now the first time | |
| 18:45 | you look at this , you're like this can't be | |
| 18:47 | geometric , it's so weird . Okay , but you | |
| 18:49 | cannot look at problems and decide if the geometric or | |
| 18:52 | not , you just can't . So what you have | |
| 18:54 | to do is calculate what this common ratio is . | |
| 18:57 | So let's go off to the side , square root | |
| 18:58 | of two , which is the second term divided by | |
| 19:00 | one square root of two . So this ratio is | |
| 19:03 | the square root of two . Now , each of | |
| 19:04 | these other ratios better be square root of two . | |
| 19:06 | Otherwise this thing is not even a geometric sequence at | |
| 19:09 | all . So let's try it . What do we | |
| 19:12 | get for the second ? It's to this term divided | |
| 19:15 | by Square root of 2 . 1st time you look | |
| 19:17 | at this , you're like well that's not equal , | |
| 19:19 | that's not the same thing . But then you remember | |
| 19:21 | , well wait a minute , we don't want radicals | |
| 19:23 | in the denominator . Let's try to get it into | |
| 19:25 | the same form . So what we can say to | |
| 19:28 | over root two , we can multiply by the conjugal | |
| 19:30 | . Remember multiply by the top and bottom by uh | |
| 19:34 | squared of two over square root of two , which | |
| 19:37 | is just multiplying by one . This is just one | |
| 19:39 | . So you haven't changed anything . So on the | |
| 19:41 | top you're gonna have let's do it down here two | |
| 19:43 | times the square root of two over . When you | |
| 19:45 | multiply these , you multiply what's under the radical ? | |
| 19:48 | So it's square of four . So what you get | |
| 19:51 | is two times a squared of two over , this | |
| 19:53 | becomes to the two is cancelled . And what you | |
| 19:56 | get is a square of two . So it turns | |
| 19:57 | out that even though it doesn't look like it actually | |
| 20:00 | when you simplify it , the common ratio does in | |
| 20:03 | fact equal square root of two . So so far | |
| 20:06 | we're pretty good . Let's check the third one just | |
| 20:08 | to make sure this guy better have the same ratio | |
| 20:12 | . So we're gonna check it two times a squared | |
| 20:14 | of two divided by two . Obviously these cancel you | |
| 20:17 | get square of two . So the common ratio here | |
| 20:20 | is again square root of two . So even though | |
| 20:22 | it doesn't look like it in the beginning , it | |
| 20:24 | has exactly the same common ratio . So it is | |
| 20:27 | a geometric sequence . Okay . And even though you | |
| 20:30 | have radicals , You do the same exact equation . | |
| 20:34 | The term is the first term times r to the | |
| 20:38 | N -1 . All right , So then the first | |
| 20:41 | term is one . The common ratio is the square | |
| 20:45 | root of two . So I wouldn't put that in | |
| 20:46 | parentheses as well to make sure not confuse it or | |
| 20:49 | kind of get confused with the radical and the power | |
| 20:51 | is in -1 . So what we're saying is that | |
| 20:54 | the term is one times this so I can drop | |
| 20:56 | that square root of two to the power then -1 | |
| 21:00 | . This is the final answer . Now , especially | |
| 21:03 | when you have a weird one like this with radicals | |
| 21:06 | . It's a really good idea to go ahead and | |
| 21:08 | try to check your work . So let's go ahead | |
| 21:11 | and at least recover the first couple of terms here | |
| 21:13 | . First term is going to be square root of | |
| 21:16 | two To the power of in -1 , but the | |
| 21:19 | term is one , so it's 1 -1 . So | |
| 21:21 | that's a zero power . So the first term is | |
| 21:24 | one . That's exactly what we have Then . The | |
| 21:27 | second term is should be the square root of two | |
| 21:31 | to the power of in -1 . But it's too | |
| 21:34 | so it's gonna be 2 -1 . That's the first | |
| 21:36 | power . So T2 is square root of two . | |
| 21:39 | That's exactly the second term . Keep going through it | |
| 21:42 | . And you'll find that the third term in the | |
| 21:43 | fourth term are easily recovered as well . And then | |
| 21:45 | you can predict any term beyond that . Notice that | |
| 21:48 | the same exact equation was used for this didn't matter | |
| 21:51 | because there was radicals or weird things going around . | |
| 21:53 | If it's a geometric sequence then it's just going to | |
| 21:56 | use the exact same equation . All right . One | |
| 22:00 | more quick one and we'll be done . The sequence | |
| 22:03 | goes as follows , 64 , negative 48 36 negative | |
| 22:10 | 27 . Now , most people including me would not | |
| 22:14 | be able to look at this and decide if it's | |
| 22:15 | a geometric sequence or not . I mean it's pretty | |
| 22:17 | easy when you're looking back at something like you know | |
| 22:20 | , something like this . Oh yeah multiplied by two | |
| 22:22 | each time . But I don't know about you . | |
| 22:25 | I'm not gonna be able to determine what the common | |
| 22:29 | multiply or what the common ratio is just by looking | |
| 22:31 | at this . So you must do the divisions in | |
| 22:34 | order to convince yourself that it is geometric . So | |
| 22:37 | let's take a look and figure out what is the | |
| 22:39 | ratio here . Take this term negative 48 over this | |
| 22:44 | term 64 . Okay . What do you get here | |
| 22:48 | ? Well I can divide the top by eight and | |
| 22:51 | divide by the bottom . I can also I can | |
| 22:52 | simplify the fraction . In other words , so divide | |
| 22:54 | the top by eight . I'm gonna get a negative | |
| 22:57 | 68 m six is 48 so negative six . And | |
| 22:59 | then divide this by eight , I'm gonna get eight | |
| 23:02 | but I see that I can simplify this further . | |
| 23:05 | I mean if I were smart I can pick the | |
| 23:06 | right factor but I didn't initially , I just knew | |
| 23:09 | I could divide by eight by eight . Then I | |
| 23:11 | look at this and say I can divide this by | |
| 23:13 | uh too . So I can say divide by two | |
| 23:16 | , I'll get negative three and divide by two , | |
| 23:18 | I'll get four . So the ratio is negative 3/4 | |
| 23:22 | . That's as fully simplified as I can get . | |
| 23:24 | Now when I checked the rest of these guys , | |
| 23:26 | I had better get negative 3/4 for each of these | |
| 23:29 | other common ratios . Otherwise this thing is not even | |
| 23:31 | a geometric sequence at all . So what do I | |
| 23:33 | have here ? I'm kind of um Running out of | |
| 23:37 | space here , I guess I will change the colours | |
| 23:39 | and I'll just go way over here , something like | |
| 23:40 | this . So the ratio is going to be 36 | |
| 23:43 | . This term divided by the negative 48 Right ? | |
| 23:48 | And I can divide these by 12 . 12 times | |
| 23:51 | three is 36 12 times four is 48 . And | |
| 23:54 | that's negative 34 It's the same as this . And | |
| 23:57 | then I can take let's change colors . Again , | |
| 24:00 | I can take these guys and find this common ratio | |
| 24:05 | as negative 27/36 negative 27/36 . And what do you | |
| 24:11 | get there ? You can divide by 99 times three | |
| 24:14 | is 27 uh and nine times four is I'm sorry | |
| 24:18 | , nine times three is 27 . So we're dividing | |
| 24:20 | by nine , so it's three and then nine times | |
| 24:22 | four is 36 so negative 3/4 same as the rest | |
| 24:24 | of these guys . So same common ratio , same | |
| 24:27 | common ratio , same common ratio . It doesn't look | |
| 24:29 | like it has the same common ratio . The reason | |
| 24:31 | it doesn't really jump out at us as much is | |
| 24:33 | because the common ratio , not only is it negative | |
| 24:35 | but the common ratio is a fraction . So what | |
| 24:37 | we're doing is we're saying if you take this and | |
| 24:39 | you multiply this by negative 3/4 you're gonna get this | |
| 24:42 | . If you take this and multiply by negative 3/4 | |
| 24:44 | you'll get this if you take this and multiply by | |
| 24:46 | negative 3/4 you'll get that now I'll leave it to | |
| 24:48 | you to verify but that's what it means . Okay | |
| 24:51 | , so how do we find the term ? It's | |
| 24:54 | just the first term . Right . The first term | |
| 24:59 | times the comment ratio Now because it's a fraction and | |
| 25:02 | because it's negative it's always a good idea to wrap | |
| 25:04 | it in parentheses And you have the power of in | |
| 25:09 | -1 . Now , the first term Is what , | |
| 25:13 | 64 ? So it's gonna be 64 multiplied by negative | |
| 25:17 | 3/4 To the end -1 , 64 . Times negative | |
| 25:24 | three forces in the end minus one . You must | |
| 25:26 | wrap this common ratio in parentheses to make sure the | |
| 25:28 | negative that the exponent applies to the negative as well | |
| 25:31 | as the fraction . And if you could just check | |
| 25:34 | right away , if you put a . N . | |
| 25:35 | Is equal to one here you get a zero . | |
| 25:37 | This to the zero is won the first tournament 64 | |
| 25:40 | which is what you expect . Put that second term | |
| 25:42 | in there and third term in fourth term , you'll | |
| 25:44 | be able to recover these just like we have been | |
| 25:46 | able to been able to for all of the previous | |
| 25:48 | problems . I leave that as an exercise for you | |
| 25:50 | . So here we have conquered the very first part | |
| 25:54 | of geometric sequences , You know , a lot of | |
| 25:56 | books just throw this thing at you and say , | |
| 25:59 | here it is guys , it's geometric sequence . And | |
| 26:01 | in a lot of students start getting in the habit | |
| 26:02 | of just memorizing it or using it , but they | |
| 26:04 | don't have any idea what they're doing . So when | |
| 26:06 | you get to a problem that is a little different | |
| 26:09 | than the basic problem , then you don't know what | |
| 26:11 | to do because you don't really understand anything . So | |
| 26:13 | what we did is we went through and talked about | |
| 26:15 | here is a geometric sequence . This is why it's | |
| 26:17 | a geometric sequence because it has a common multiplier , | |
| 26:20 | which is also the exact same thing as a common | |
| 26:22 | ratio . Because multiplication and division are closely related . | |
| 26:25 | So , common ratio , common multiplier , same kind | |
| 26:28 | of thing . That's how you recover the terms . | |
| 26:29 | And then we basically said if you take a common | |
| 26:32 | like 10 terms for instance , then you have nine | |
| 26:34 | ratios . So to recover that 10th term , it's | |
| 26:37 | just the first term times the common ratio , nine | |
| 26:40 | times which ends up recovering what that equation is . | |
| 26:45 | So that's really what I want you to pull out | |
| 26:46 | of it . The problems are great , but I | |
| 26:48 | really want you to understand what the equation means . | |
| 26:50 | Solve every one of these problems yourself , make sure | |
| 26:52 | you're getting the correct answer . Follow me on to | |
| 26:54 | the next lesson . We're gonna work some slightly more | |
| 26:56 | challenging problems with geometric sequences . |
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