16 - The Geometric Series - Definition, Meaning & Examples - Part 1 - Free Educational videos for Students in K-12 | Lumos Learning

16 - The Geometric Series - Definition, Meaning & Examples - Part 1 - Free Educational videos for Students in k-12


16 - The Geometric Series - Definition, Meaning & Examples - Part 1 - By Math and Science



Transcript
00:00 Hello . Welcome back here . We're going to conquer
00:02 the topic of the geometric series . Now of course
00:04 we've already talked about the geometric series in several lessons
00:07 but we never calculated the some of the terms of
00:11 the geometric series . So just like in the last
00:13 lesson we talked about the arithmetic series , we introduce
00:16 the concept of the arithmetic series . We talked about
00:18 the some , we wrote the equation down and I
00:20 proved it to you and then we solve problems .
00:22 We're gonna do the exact same thing for the geometric
00:24 series . I'm gonna introduce it . I'm gonna motivate
00:26 it . I'm gonna write down the equation for the
00:28 some of the terms of the geometric series . Then
00:31 I'm going to prove it to you mathematically . Then
00:33 we'll solve some problems . So let's talk about the
00:35 simplest geometric series that I can think of uh to
00:39 motivate the point here . So this is a geometric
00:41 series . How about one plus two plus four plus
00:46 eight plus 16 plus 32 Plus 64 . So I'm
00:52 just trying to pick something that's easy to understand with
00:55 a finite numbers . How many terms do we have
00:57 ? 1234567 terms . So there's seven terms . Uh
01:01 and how do we know ? It's geometric geometric .
01:03 Just means that to get the next term you take
01:05 the previous term , you multiply by the same number
01:08 by a common ratio . So to go from this
01:10 term to this term , we actually multiply by two
01:13 to go from this term to this term . We
01:15 multiply by two to go from this term to this
01:17 term , multiplied by two . This term to this
01:19 term multiplied by two . And I can keep writing
01:21 it . You can see multiplied by two , multiplied
01:23 by two . So this is a geometric series and
01:26 the common ratio R is equal to two . This
01:28 is the common ratio . All right . So ,
01:34 what I want to do is I'm gonna write down
01:36 the some of these terms . And when we've written
01:38 these things down , we've done the summation convention ,
01:40 but we haven't actually calculated the sum of any of
01:44 these geometric series like this . So let's go and
01:46 write that down . The sun of the geometric series
01:57 is as follows . The partial sum , which means
02:01 we sum the first in terms is equal to the
02:04 first term multiply by one minus R to the power
02:07 of n . All divided by one minus r .
02:12 Okay , that's what I want you to remember now
02:14 . Is it obvious that this is the case ?
02:15 No , but we're going to prove it in just
02:17 a second . Now , the one thing I want
02:19 you to uh to point out to you here is
02:21 that the only thing you need to calculate the sum
02:24 of these terms . I mean , I know you
02:25 can add them in your calculator . What I'm saying
02:26 ? What if you had 1000 terms that would take
02:27 forever ? How do you calculate the sum ? Well
02:30 you put the number of terms in place . You
02:32 have to go in , you have to know the
02:34 common ratio are , that's what you're multiplying by to
02:36 get these terms are is down here as well and
02:39 you have to know the first term . So you
02:40 actually don't need to know the last term in order
02:43 to do this . Remember for the arithmetic series we
02:45 needed the first term , We needed the last term
02:48 and we need the number of terms here . We
02:50 do not need to know the last term . We
02:52 just need to know what is the common ratio .
02:54 What is the number of terms ? What is the
02:56 first term ? Now , notice in the denominator we
03:00 have one minus R . And because we have that
03:02 in the denominator , what we really need to stay
03:04 out here is that this equation holds for the sum
03:07 . However are cannot be equal to one . Why
03:10 ? Because if I put a one in for the
03:12 common ratio one minus one is zero and we're dividing
03:15 by zero , that's really undefined . Or you could
03:17 say that it goes off to infinity . So either
03:19 way , when you divide by zero it's not going
03:21 to work for a calculation . So you cannot have
03:23 our common ratio equal to one . But when you
03:26 think about that's not a big deal because if I
03:28 had a common ratio of one what would the series
03:30 look like if I take this and multiply by one
03:32 , I'm gonna get one multiplied by one . Again
03:34 I get one multiplied by uh one again I get
03:37 one . So if the common ratio wherever one for
03:40 any series technically it's geometric but it's a boring series
03:44 because every term is just going to be the exact
03:47 same term . So if I wanted to find the
03:49 some of the terms of a geometric series with a
03:51 common ratio of one , I don't need this fancy
03:54 equation . If all of the terms of the same
03:56 , I just take the first term and multiply by
03:58 how many terms I have . If the first term
04:00 is five then every term is five and I just
04:03 say five times however many terms I have and then
04:06 I have the some so it's kind of boring and
04:08 it's not it's not really helpful . And so it
04:11 isn't a big deal that are cannot be won in
04:13 this equation for any other geometric series . This equation
04:16 is going to work now before we actually prove this
04:20 , I want to mathematically prove it to you .
04:22 Before we actually do that , I want to apply
04:24 this to this known geometric series that we have above
04:28 . So we just said that the partial sum is
04:32 the first term times one minus R to the power
04:35 , then over one minus are . That's what we
04:38 said , and we said we have 1234567 terms .
04:42 So for this particular series we know that N is
04:45 equal to seven , We know that the first term
04:48 is equal to one and we know that the common
04:51 ratio , because we've been we just wrote it down
04:52 here . The common ratio is too , so we
04:54 have everything needed . So let's check it out .
04:56 The seventh , partial some seven partial sum is the
04:59 sum of the first seven terms is going to be
05:02 equal to this first term , which is one times
05:05 one minus the common ratio of our two to the
05:08 power of n however many terms we're adding together and
05:11 on the bottom it's one minus are so one minus
05:13 R . Is to all we have to do is
05:15 do this calculation . Okay so the one times this
05:20 the one is going to disappear . So all you're
05:21 going to have on the top is one minus what
05:24 is to do the seventh power you're gonna have 128
05:27 Wrap it in parentheses if you want on the bottom
05:30 1 -2 is gonna be negative one . So what
05:33 are you gonna get ? You're gonna have s seven
05:36 you're going to get one minus 1 . 28 is
05:38 negative 1 27 over negative one . So the seventh
05:43 partial sum is positive 127 . So what we're basically
05:47 saying is that if we add up all of these
05:49 terms we should get 127 and I encourage you to
05:52 do that , grab a calculator and say one plus
05:55 two plus four plus eight plus 16 plus 32 plus
05:57 64 . You're gonna figure out that that's exactly equal
06:00 to 127 . Now for a series with seven terms
06:04 , it's not a big deal to add them up
06:06 . But what if this geometric series had 1000 terms
06:08 ? What have had a million terms or 15 million
06:10 terms and adding them up ? Even with the computer
06:13 would take forever . Right ? I take a long
06:14 time even with the computer . So this equation is
06:18 mathematically calculating what the sum of however many terms of
06:22 the series I want . All I need to know
06:24 is the first term , the common ratio which lets
06:27 me predict all the future terms and however many terms
06:30 I want to consider , it's gonna add them up
06:32 now . The question is , why does it work
06:34 right now , As I said in the last lesson
06:36 , when we derive the arithmetic series formula in math
06:39 , When you prove things , all you care about
06:41 is arriving at this . You want to prove that
06:43 this is true and you can do anything you want
06:45 legally to get there . So what we're gonna do
06:48 is we're gonna write down the terms of the series
06:51 . We're gonna , in this case this is a
06:52 geometric series I wrote down with numbers , but we're
06:54 going to write down a general geometric series . And
06:57 then I'll walk you through the next steps to get
06:59 from point a down to proving that this is the
07:02 equation that describes the son . Mostly I don't want
07:05 you to memorize this , but mostly I just want
07:07 you to know that these equations don't come out of
07:09 thin air , Right ? So how do we do
07:10 it ? Let's take a look at this , proof
07:13 . What I'm going to suggest . What I'm gonna
07:14 say is that the inthe partial sum is gonna be
07:17 the sum of all the terms of the geometric series
07:19 . Every series starts with the first term . So
07:22 we call it T one . What is the next
07:24 term in this series ? We'll for a geometric series
07:26 to get the next term . All you do is
07:28 you start with the first term and you multiply by
07:30 a common ratio to get the next term after that
07:32 . And you multiply by the common ratio , it's
07:34 always the same pattern . So to get the next
07:36 term , it's just gonna be that first term times
07:38 the common ratio . RT one times are . But
07:42 then how do you get the next term after that
07:44 ? It's gonna be this term times are but then
07:47 times are again , so it's going to end up
07:48 being T1 times r squared when you think about it
07:52 , you multiply by two and you multiply by two
07:54 again . So basically you consider it being one times
07:57 four to give me four . So I can predict
07:59 any future term by doing this . So this is
08:02 the pattern like this . And then let me ask
08:04 you another question . Uh Well let me let me
08:07 go and get it down . What would the last
08:09 term be in the series in the last term ?
08:11 Here we had noticed we had seven terms 1234567 There's
08:16 seven terms . How many multiplication is do I need
08:18 to get their one multiplication ? 23456 multiplication . So
08:23 if I start with the first one and do six
08:25 multiplication is I get to the seventh term . So
08:28 in the general form the last term is going to
08:31 be the first term Times are to the power of
08:35 N -1 . So we start with the first term
08:37 to get the next term multiplied by the common ratio
08:40 . Then we multiply by the common ratio again .
08:42 And I'm saying we keep doing that until we get
08:43 to the last term . But the last term is
08:46 R to the n minus one . Why is it
08:48 to the n minus one for the last term ?
08:49 Why isn't it just are to the 10th ? Well
08:52 it's because if they're in terms in the series If
08:55 there's seven terms in this series I do six multiplication
08:59 is to get there . So this is the last
09:01 term , the 10th term . So I do one
09:03 less multiplication is by our to get there . That's
09:05 why it's in -1 . What would the term just
09:08 before that be ? It would be T one Me
09:13 erase This plus make it clear T one R to
09:17 the N -2 . Because again we're multiplying here .
09:21 So this is multiplied by arts and M -1 .
09:24 And this is one less multiplication . Uh going that
09:27 direction . Okay . And then we have dot dot
09:30 dot in the center here . I wish I wouldn't
09:32 smudge that right there . That's okay . We'll get
09:35 there . So this should hopefully make sense . Especially
09:38 when you think about the numbers here , when you
09:41 think about the numbers , you multiply by two by
09:42 two by two and then eventually have a last term
09:45 multiplied In -1 times . And so that's exactly what
09:48 we had here . Last term in -1 times in
09:51 -2 times . And then we fill everything else in
09:54 . All right now , what I want to do
09:56 is I want to multiply this series by are the
09:59 common ratio are and I'm gonna write that down directly
10:02 underneath it and then I'm going to subtract it .
10:04 Why are we doing this ? You can't predict ahead
10:06 of time . This has been proved a long time
10:08 ago by people that played around with it and figured
10:10 it out that it works so we're going to do
10:12 it . But don't worry so much if you don't
10:14 , if you say well I would never would have
10:15 figured that out . That's okay . I didn't wake
10:18 up this morning and decided that I was going to
10:20 do this either . All I care about is that
10:21 you understand what I'm doing . So let's take this
10:24 exact same series and multiply by are the common ratio
10:27 are and I'm gonna change colours to do that .
10:30 The common ratio our is our times as cement .
10:35 So I multiply the left hand side by our I
10:37 must also multiply the right hand side by our The
10:40 first term is this , I'm gonna multiply by our
10:43 but I'm not gonna write it underneath it . I'm
10:45 gonna I'm gonna multiply every term ir but I'm going
10:47 to shift the terms over so I'm gonna multiply by
10:49 this and I'm gonna write it down here . T1
10:52 times are why am I not writing it down here
10:55 ? Because ultimately I'm going to be adding these terms
10:57 together and these terms go together . I could write
10:59 it here and then I would just have to collect
11:01 terms but it's gonna be easier if I line that
11:03 turns up . So I multiplied by our I write
11:05 it here , I multiply by our here and I
11:07 write it underneath here . T one r squared .
11:10 I multiply this by our and I'm gonna get our
11:12 cube but that's gonna fall into the dot dot dot
11:15 area so I'm not going to have to actually write
11:17 that . It is there , it's just not written
11:19 down , it's in the dot dot dot area .
11:21 And then I'm going to multiply the term right before
11:24 this one by R . C . This will be
11:26 in minus three right here , multiply by . Are
11:29 you add the exponents ? It'll be T one R
11:31 to the n minus two because this one in minus
11:34 three Plus one from the art , you're going to
11:37 get in -2 . Then when I multiply this one
11:40 by our I'm gonna have our multiply all this stuff
11:43 by our but I'm gonna add the exponents , it'll
11:45 be a -1 here , so it'll be T1 are
11:49 in -1 . And then when I multiply by our
11:51 to the last term are to the first power will
11:54 add with this so to become our to the 10th
11:56 power . Um Actually I forgot A . T .
12:00 It will be T . One R . To the
12:04 power . So all I've done is multiplied by this
12:06 by our write it down . Multiply this by our
12:09 write it down here , multiply this by our write
12:11 it down by here , multiply this by our it's
12:13 in the dot dot dots . Multiply the term before
12:16 here by our it's going to be in minus two
12:18 . When you add the exponents multiply this by are
12:21 you subtract add one to this ? It will be
12:23 in minus one , multiply this by our , add
12:25 the exponents . It'll be our to the end .
12:27 So you know it's legal to multiply any equation by
12:30 whatever you want , as long as you do it
12:31 to the left and the right hand side . So
12:33 I've done that now that I've done that . What
12:36 do I want to do ? I'm going to take
12:38 this equation and I'm going to subtract it from the
12:41 one right underneath . I'm gonna do a subtraction .
12:44 Remember when we solve systems of equations ? We can
12:47 add them or subtract them ? That's fine . Okay
12:50 , so what am I going to get sn minus
12:52 ? This is just gonna be S N minus R
12:56 S N . Just this term minus this term .
12:58 No magic here , what's this gonna be ? T
13:00 one minus zero is gonna give me T one .
13:04 What is this term minus this term ? What's exactly
13:07 the same terms ? So it's zero . What does
13:09 this term minus this term ? It's also zero .
13:12 And all the terms in the dot dot dot areas
13:13 are also going to give me zero . So let's
13:15 just kind of do this . This term minus this
13:18 term is exactly the same thing . It's gonna be
13:19 zero . This term minus this term is exactly the
13:22 same thing is going to be zero . Here's a
13:24 zero up here minus this one has to be a
13:26 negative T . One R to the end because it's
13:29 zero minus this because I'm subtracting everything so really everything
13:33 cancelled and the whole thing except for all of this
13:35 stuff . So I have S . N minus R
13:38 . S . And equals T . One . All
13:41 of this is zero minus this T one R to
13:45 the end . So now all I have to do
13:48 is clean up , I have an S in here
13:51 So I pull it out , factor it out .
13:52 I get 1 - are on the left 1 -2
13:55 . On the right . I pull out a .
13:57 T . One , I'm gonna have one minus R
14:00 . To the N . To verify . You multiply
14:01 through here you get this , you multiply through here
14:04 . You get this . Now we're going to solve
14:05 for the impartial some it's gonna be T . 11
14:09 minus R . To the power of N . Divided
14:11 by this whole thing one minus R . Which is
14:15 exactly what we said . The partial sum should be
14:18 the first term times one minus R . To the
14:20 power then divided by one minus R . And of
14:23 course because this is in the bottom Our cannot be
14:26 equal to one . We've already discussed why ? That's
14:29 not that big of a deal anyway because if the
14:31 common ratio were won the series would be really simple
14:33 . We could easily add everything together . Now do
14:36 I expect you to be able to recover or do
14:38 a proof like this ? No of course not .
14:40 I don't care about that . I just want you
14:42 to know that . You understand where it comes from
14:44 . Okay it's legal . Every step is legal .
14:46 We wrote the series down . That's legal . We
14:49 multiply this series by our which is a number .
14:52 It's legal to do that . As long as we
14:54 multiply the both sides of the equation by the same
14:56 number . Legal , it's totally legal to subtract two
14:59 equations of one another , you know that ? So
15:01 we subtract it and then when you manipulate terms ,
15:03 you're able to get an expression for the sum and
15:05 this is the sum that's presented in the theorems .
15:08 Alright , so um put it in your back pocket
15:11 , make sure you understand where it comes from .
15:13 But mostly we want to focus on from here ,
15:14 going on how to apply it . So let's go
15:17 to this next board and just do a couple of
15:19 quick extra problems to apply it . What if I
15:23 have a geometric series within terms We're in his eight
15:28 . And the common ratio between these terms is to
15:31 and the first term is equal to one . And
15:34 I ask you to find the sum . The first
15:36 thing you do is you say well the impartial some
15:39 is you should write this equation down every single time
15:41 one minus are to the end Over 1 -2 .
15:45 Now there's eight terms . So what I want is
15:47 I want the eighth partial sum is the first term
15:51 which is a one Times This 1 - are but
15:55 that's too to the power of in but that's eight
15:58 . There's eight terms here Over 1 -2 . But
16:01 again ours too . So it's 1 -2 . So
16:04 the 8th partial sum is The one is just gonna
16:07 multiply out . So what do you have here ?
16:09 You have 1 -2 to the eighth power . When
16:12 you take two to the power of eight you get
16:14 256 On the bottom you get -1 . So here
16:19 you have negative 255 over -1 . So The 8th
16:24 partial sum is positive 255 . That is the final
16:28 answer . Now . I always recommend when we do
16:32 these problems not to just plug things in and go
16:35 on a merry way . Let's write the series down
16:38 just so we know where it comes from . Since
16:41 we know the first term , we can write the
16:43 first term right away . It's one and we know
16:46 the common ratio is too and we know there's eight
16:48 terms total . So what we're gonna do is multiply
16:50 by two because that's the common ratio will give it
16:52 to multiply by two . Again we'll get a four
16:55 . Then times two is eight . Times two is
16:58 16 . Times two is 32 then times two is
17:01 64 . Times two is 128 . We have 12345678
17:07 terms , which is what we said , we were
17:09 gonna add up . So this is eight terms .
17:13 So it's basically the same series is what we started
17:15 out with , but we just have one extra term
17:17 and if you grab a calculator and add these numbers
17:19 up , you will find out that you get 255
17:22 . And so that's what we're gonna do . So
17:25 I only have one more of these guys to illustrate
17:27 how to use the geometric series equation . Mostly I
17:31 wanted to introduce it . I wanted you to understand
17:33 where it comes from , but the proof and then
17:36 solve some problems . What if I ask you solve
17:38 and give me the some of the following series ,
17:41 two minus six plus 18 minus 54 plus 160 to
17:49 minus 486 Plus 145 , 8 minus 43 74 plus
18:00 13 +12 to minus 39 +366 How many terms do
18:06 I have ? +123456789 10 terms . Let me just
18:11 check to minus six plus 18 minus 54 plus 1
18:15 60 to minus 46 plus 14 58 plus . Uh
18:19 Oh I missed one . Let's see . 486 Plus
18:24 1458 . Oh no I didn't minus 4374 plus 1312
18:30 to minus 39366 That's correct . So what we want
18:34 to do is we want to find the some of
18:35 this first of all , is it a geometric series
18:37 ? That's what we want to figure out . Well
18:39 if you think about it , you grab a calculator
18:41 . If you multiply by negative 32 times negative three
18:45 , you'll get the negative six if you take this
18:47 and also multiplied by negative three . Six times negative
18:51 three , you're gonna get the 18 . If you
18:53 go here and multiply by negative three , you'll get
18:55 this multiplied by negative three , you'll get this multiplied
18:57 by negative three , you'll get this and you go
18:58 all the way through the sequence . So every term
19:01 is just the previous term multiplied by negative three .
19:03 So we then know that it's geometric with the common
19:06 ratio of negative three . We also know that they're
19:10 in equals 10 terms because when you count the terms
19:13 you get 10 and we also know that the first
19:15 term is equal to two . So by giving being
19:18 given the sequence we can pull all of this information
19:21 out and that's all that we need to solve this
19:22 problem . Okay , The 10th partial sum is equal
19:28 to the first term , times one minus R .
19:32 To the power of N divided by one minus are
19:35 . So if we want to find , how many
19:36 times do we have 10 ? The 10th partial sum
19:39 , it's going to be the first term which is
19:40 two times one minus the common ratio . Are are
19:44 we said was negative three . So you put in
19:46 parentheses negative three . Make sure you put in parentheses
19:49 to the power of N , which is 10 .
19:52 And on the bottom it's one minus R . But
19:54 again , Rapidan princes because R is negative , it's
19:56 a negative three . So now we just have to
19:59 crank through this . What do we get ? Two
20:00 times ? Here's one right here minus what is negative
20:04 ? Three to the 10th , negative three to the
20:06 10th is going to be a positive number 59 oh
20:10 49 It's a big number 5 59,000 and 49 .
20:13 But it's positive because it's a an even power the
20:17 minus sign comes from the outside here . This evaluates
20:20 to a positive number here . You have won .
20:23 This becomes a plus three . So you have a
20:25 four . All right . So the rest of it
20:28 is just simply math . The 10th partial sum is
20:32 two times when you do one minus this , you'll
20:34 get negative 59 048 over four . So the 10th
20:41 partial sum is negative . 29,000 524 negative . 29
20:46 5 24 you might say wait , wait , wait
20:48 . How's it negative ? Well , it's negative because
20:50 the signs are alternating and look at the last number
20:53 . The last number was 39,000 and 366 . That's
20:56 a negative number . This is going to pull the
20:58 whole thing negative because all the other numbers are so
20:59 much smaller . And so you end up with negative
21:01 29,524 . So in this lesson we have uh used
21:08 the geometric series to uh the equation for the geometric
21:11 series to calculate the sum of any geometric series we
21:14 want and all we have to know to do that
21:16 . We do not need to know the last term
21:18 , but we do need to know the first term
21:20 , whatever that common ratio is and however many terms
21:22 we have in the series we went through the proof
21:25 of it , the proof of it . I hope
21:26 you understand that . I try to make it as
21:28 clear as they can . The proof is important ,
21:30 but mostly I want you to know how to use
21:32 it and that's why we did these problems here .
21:34 So make sure you can solve every one of these
21:36 yourself . If one more lesson in geometric series calculations
21:39 that we're gonna do , so follow me on to
21:41 the next lesson , we'll get more practice with the
21:43 geometric series .
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