07 - The Geometric Sequence - Definition & Meaning - Part 1 - Free Educational videos for Students in K-12 | Lumos Learning

07 - The Geometric Sequence - Definition & Meaning - Part 1 - Free Educational videos for Students in k-12


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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