The Arithmetic Series - Part 1 - [14] - Free Educational videos for Students in K-12 | Lumos Learning

The Arithmetic Series - Part 1 - [14] - Free Educational videos for Students in k-12


The Arithmetic Series - Part 1 - [14] - By Math and Science



Transcript
00:00 Welcome back to this lesson . The title is called
00:03 the arithmetic series . This is part one and we
00:06 have a part to with additional problems to follow on
00:08 with this guy right here . So up until this
00:10 point we've concentrated on what is the concept of a
00:14 series in general when you add up terms . And
00:16 we've also concentrated on the sigma notation with the big
00:19 greek letter sigma . That's how we write down when
00:22 we add up a sum of terms . So that's
00:24 in general how we're going to represent all of these
00:26 uh what we call series when we add up a
00:28 bunch of things . Now , we've also talked about
00:30 a few special , very special series in passing .
00:32 We've talked about the arithmetic series and we talked about
00:35 the geometric series but we have never actually added up
00:38 the terms of the arithmetic series . And we've never
00:40 actually added up the terms of the geometric series .
00:42 So that's what we're gonna concentrate on here . We're
00:44 gonna go back and revisit the arithmetic series . We're
00:47 right down the terms are going to talk about it
00:49 . And we're going to actually figure out an equation
00:51 or formula to add up the terms of an arithmetic
00:55 series . And I'm gonna prove that this equation works
00:58 . I'm going to do a proof to show you
00:59 how it works . We're gonna apply to some problems
01:01 and in a few lessons will do the exact same
01:03 thing for the geometric series . So at the end
01:05 of these guys , you should understand what arithmetic series
01:08 is , what a geometric series is , how to
01:10 find their some without actually grabbing a calculator and adding
01:13 the terms up . We want to be able to
01:15 , for instance , you might have 1000 terms in
01:16 your series . You want to be able to find
01:18 some without actually adding up all of those numbers in
01:20 the calculator . So that's what we're gonna be doing
01:22 here and this kind of thing goes on and on
01:24 and used in algebra and trig and calculus and beyond
01:27 . So it's something that you're going to see over
01:28 and over again . So we want to talk about
01:30 the concept of an arithmetic series . So let's go
01:33 back and let's pick a very simple one so that
01:36 we can talk about something concrete . Let's talk about
01:38 the arithmetic series that looks like this two plus four
01:42 plus six plus eight plus dot dot dot plus 100
01:47 . Now , how do you know this is an
01:49 arithmetic series because arithmetic sequences which are just the listing
01:53 of the numbers , they always differ by a constant
01:56 by a number that we just add to each of
01:58 the terms to get the next term in the sequence
02:00 or in the series in this case . So here
02:01 we're adding to that , we're adding to that ,
02:04 we're adding two and so on . And you go
02:05 all the way up to your final term being 100
02:07 . So because we know this then we know that
02:10 this is arithmetic a riff medic and it has to
02:16 be true for all of the terms every term you
02:17 must be able to get by adding a number to
02:20 the previous term . But what we want to do
02:21 is figure out what is the sum of this arithmetic
02:24 series . Now obviously I can grab a calculator and
02:27 go to plus four plus six plus eight plus 10
02:29 plus 12 . Ah bah bah bah bah all the
02:31 way till I get to 100 . It's going to
02:33 be a big number and I'm gonna be pressing plus
02:35 a bunch of times . We don't want to do
02:37 that . We want to figure out what is an
02:38 equation that will let me calculate the sum without actually
02:41 adding up all of those terms . So , in
02:44 order to understand how to do that , we need
02:46 to talk about , I need to introduce a term
02:48 . And this term that we're gonna talk about is
02:49 very simple . It's called the partial sum . So
02:52 let's look at this series . This is term number
02:54 one , term , number 234 and so on and
02:57 so on . And then we have eventually we're gonna
02:59 have 50 terms when you really look at it all
03:01 because in 2468 10 and then all the way to
03:05 100 is actually gonna be 50 terms in this series
03:08 . So we have the concept of a partial some
03:15 right , what is a partial sum ? We use
03:17 the letter S to denote . That's a kind of
03:20 weird looking s sorry about that , but we use
03:21 the letter S to denote partial sum . So if
03:25 I'm going to talk about the partial some of the
03:27 first two terms I put in number two under the
03:29 S . That means I just want to add up
03:31 the first two terms . So what is this gonna
03:33 be ? It's gonna be two plus four . I
03:35 know that you can add that up and you'll get
03:36 six . I don't care about calculating to some .
03:38 Now . I just want you to know that when
03:39 I put an S . With a two underneath it
03:41 just means add the first two terms of the series
03:43 together . Okay if I put S with a little
03:47 three under it it's two plus four plus six .
03:50 This is the partial some of the first three terms
03:53 of this arithmetic series . And you might guess that
03:55 the S sub four is two plus four plus six
03:59 plus eight . Okay , the first four terms of
04:03 this arithmetic series . So this is the concept of
04:05 a partial sum . So why am I writing down
04:07 partial sum ? Because now we want to generalize the
04:10 partial sum to a general equation . What I'm gonna
04:13 do here is I'm just gonna write down the answer
04:15 . I'm gonna write down what the sum of any
04:17 arithmetic series is . Right . Then we're gonna do
04:20 one quick little problem and then I'm going to prove
04:22 it to you with a formal proof to show you
04:24 why this equation I'm going to write on the board
04:26 actually works as I want you to know where things
04:28 come from right . And then we'll work some more
04:30 problems at the end . So these are the partial
04:32 sums , first two terms in the first three terms
04:34 in the first four terms . And then we have
04:37 what we call in general the some of any arithmetic
04:42 , not just this one , but any arithmetic series
04:48 and that some is the following instead of first two
04:51 terms . First three terms . 1st 1st 4 terms
04:53 . It will be the first in terms however many
04:55 terms I want to add together is going to be
04:57 equal to end , multiplied by the first term plus
05:02 the last term that I care to have in my
05:05 in my edition , divided by two . So this
05:08 is what you will see in any textbook in algebra
05:11 . Pre calculus , calculus , things like this .
05:13 Okay , so what it's telling you is if I
05:16 want to figure out for instance the first four ,
05:19 the partial sum , the first four terms that would
05:22 put a four in here , then the equation would
05:24 be four multiplied by the first term , two plus
05:28 the last term tisa before because remember I put four
05:30 in here , the fourth term which is eight .
05:33 I would add those together , multiplied by in which
05:35 is four . And then I would divide by two
05:37 . So this allows me to use this arithmetic series
05:41 formula to find the sum of any arithmetic series I
05:44 want . For instance in this one here I have
05:46 a lot more than four terms I have . It
05:48 goes all the way to 100 but I'm skipping by
05:50 two . So there's actually 50 terms there . So
05:53 for that I would put 50 in here . 50
05:55 in here , I have the first and last terms
05:57 and then I would divide by two . Now before
05:59 we go off and actually do it I want you
06:01 to kind of understand what this equation really means .
06:04 So I'm gonna write it again . We're gonna say
06:06 s sub in for the end number . The partial
06:08 the partial sum is the way you would say this
06:10 is equal to end . Instead of writing it like
06:13 this , I'm gonna write it like this and multiplied
06:15 by a fraction which will be T . One plus
06:18 T in the first and the last term divided by
06:21 two . All I've done is kind of break it
06:23 up so it's not written like this and pull the
06:25 in out , multiplied by this guy . Right here
06:29 , what is this term ? What is it ?
06:32 What is that term ? This term is the average
06:37 value of all terms . All terms in the seat
06:45 in the series that I'm considering , cause I'm trying
06:47 to only add up to the term . So if
06:49 I take the first term plus the intern the last
06:52 one that I'm considering and I divide by two then
06:54 this is just the average value of those terms .
06:56 Right ? And then of course I'm multiplying by in
06:59 terms and I'm hoping that you can see by breaking
07:03 it out like this and talking about what this equation
07:05 for the some of the arithmetic series is really doing
07:08 okay . What's going on is when you're adding up
07:10 terms , you have a first term and then you
07:13 have the next term , next term , next term
07:14 , and then you have some last term over here
07:16 . But the arithmetic series , every term just differs
07:20 by adding a number , adding a number , adding
07:22 a number . So if I take the first and
07:24 last terms adam together , divide by two , I'm
07:27 going to get the average value . I'm gonna get
07:29 the value right in the middle of all of those
07:31 terms . Right ? And by using that average value
07:33 and multiplying by how many terms I have , I
07:36 can arrive at the at the exact value of the
07:38 some . That's what it's basically doing . I want
07:41 to illustrate a little bit more why it works over
07:45 here . I think I want to do it on
07:47 the next board over here . So let's revisit our
07:50 series here . Let's say we have two plus four
07:53 plus six plus eight plus dot dot dot plus 100
07:58 . I want to add these up . It's not
08:00 100 terms because I'm skip counting . Right ? So
08:02 it's actually 50 terms . So what I'm gonna write
08:04 down is the fact that What this arithmetic series Here
08:09 is . It's 50 terms , 50 terms . And
08:15 also the first term is equal to two , right
08:21 ? And the 50th Term When you go and count
08:24 throughout the 50th term is 100 . That's the last
08:27 term in the series that I'm considering , right .
08:29 And I want to figure out what is the partial
08:32 some of the 1st 50 terms . See , this
08:34 is the this is the partial some of the first
08:36 two terms . The partial some of the first three
08:38 terms . The partial some of the first four terms
08:40 . I want to find The partial some of the
08:42 some of the all 50 of them . So according
08:44 to this equation , the way it's gonna work is
08:46 you're going to come down here and say , well
08:48 it's in times T one plus the first and last
08:52 terms divided by two . So I have how many
08:54 terms ? I have 50 . Right , And then
08:57 I have . The first term is to The last
09:00 term is 50 . I'm sorry , not the last
09:02 time was 50 . Last term is 100 100 And
09:06 it's divided by two . Right ? So what do
09:09 I get here when I do this ? S 50
09:12 It's going to be 50 multiply . What is this
09:14 ? 102 divided by two is gonna give you 51
09:18 . Okay , what is this ? 51 ? This
09:22 51 is the average value of all terms . In
09:30 other words , I have so many terms here ,
09:32 I can't really write them all out but imagine I
09:34 wrote them all out in a line Then somewhere right
09:37 in the middle , literally in the middle of all
09:39 of those terms would be the number 51 that represents
09:41 the average value of all the terms . Now I
09:44 want to find the sum of all of these terms
09:46 terms . Some of these terms are smaller than the
09:49 average , like all of these are smaller than 51
09:53 But an equal amount of them are gonna be bigger
09:55 than 51 , going all the way to 100 .
09:57 So even though some of these terms are smaller and
10:00 so when I add them up it's smaller than this
10:02 average number . But there's an equal number of terms
10:04 that are bigger than 51 that are too big .
10:06 So it all balances out so that if I just
10:09 take this and multiply by the number of terms ,
10:12 have . even though some of the terms are smaller
10:14 and some of the terms are larger on average by
10:17 taking the average value multiplied by the terms I'm getting
10:20 the actual some and it's always going to work out
10:22 that way . And the only reason it works is
10:24 because arithmetic series with arithmetic sequence numbers , they're all
10:29 evenly spaced like that , right ? Everything is evenly
10:32 spaced . So by finding an average value in the
10:34 middle and knowing how many terms I have and multiplying
10:36 I'm going to get the actual some of everything against
10:39 some of the terms will be too small and contribute
10:41 less to the some . But the terms on the
10:43 other side of this number are going to be too
10:45 big and they overcompensate toward the some . It'll all
10:47 balance out so that the actual some is gonna exactly
10:50 equal what this to what this is . So when
10:53 you do this in the calculator , uh the sum
10:56 of all 50 of those terms is going to be
10:59 2550 . And that's the number that you would get
11:04 if you literally sat down . And in the calculator
11:06 22 plus four plus six plus eight plus 10 plus
11:08 12 plus 14 plus 16 . All the way up
11:10 till plus 100 . All right , So this is
11:13 essentially how we're going to use this equation . We're
11:16 gonna use the arithmetic sequence equation . I'm sorry ,
11:19 the arithmetic series equation to calculate the sum without actually
11:24 adding up all the terms . But you might ask
11:26 yourself why does it work now ? I've tried to
11:27 kind of explain by using averages , give you an
11:30 actual feel for why it works like in your mind
11:33 the whole argument I've just done hopefully has helped you
11:35 but we still want to do a proof . Even
11:38 if you don't totally understand the proof or just kind
11:40 of get the basic idea . It's still a really
11:42 good idea for you to watch it because it gives
11:44 you a feel for how things are approved in math
11:47 and math . All I care about is figuring out
11:49 what the equation is for the partial sum the impartial
11:53 some I can do anything legal that is possible to
11:56 do to get there . So what we're gonna do
11:58 is we're gonna write a generic series down . In
12:01 other words , we I picked numbers to make it
12:02 easier to understand but we're gonna write down a general
12:05 series and then we're gonna play around with it and
12:07 you'll see how at the end of the day we're
12:09 gonna get this equation straight out of the proof .
12:11 So let's go and do that over here on the
12:13 other board . Right . First thing I want to
12:16 do is I want to write down The I want
12:19 to write down the series two times . And the
12:22 first time I'm going to write it down the regular
12:23 way and then I'm gonna kind of walk you through
12:25 what we're gonna do after that . So the inthe
12:28 partial sum is going to be a bunch of terms
12:31 that we're going to add together . What are those
12:32 terms going to be ? It's going to be the
12:34 first term that's always t sub one plus . This
12:37 is an arithmetic series . So every term just differs
12:41 from the previous term by adding a number . That
12:43 number we called . The common difference D . In
12:46 this series , the common difference was two we add
12:48 to we had to we had to so the common
12:50 difference was too . So for the second term in
12:53 this generic series it's just gonna be T one plus
12:55 two . I'm sorry not T one plus two because
12:57 it's a general we're trying to keep it general it's
12:59 gonna be T . One plus D . D .
13:02 Is just a common difference in our example before D
13:05 was to what would be the next term after this
13:08 ? Well what I would do is that would add
13:09 D . Again . So ultimately it would be T
13:11 one plus two times . D think about it because
13:15 you go here this is plus D . These two
13:18 in this case this is plus two again . So
13:20 to go from here to here I'm just adding four
13:22 . So to go from here to here I'm just
13:24 starting with the first tournament adding two times the common
13:26 difference so I can keep going down this way .
13:29 But ultimately I'm gonna put a dot dot dot and
13:32 I'm gonna start toward the end . Okay . The
13:35 very last term of the series is gonna be T
13:38 seven , that's the very last term . The first
13:39 term is T one . The very last term is
13:42 tien what term comes right before TC event . Well
13:45 it's going to be um T seven -D . Right
13:51 , think about it . If the last term ,
13:52 if you go back to our numbers , if the
13:53 last term was 100 then the term right before it
13:56 is 100 minus two which is 98 . So all
13:59 I'm doing is I'm saying the last term is T
14:01 . Seven and the one right before it is T
14:04 . Seven minus D . So let me clean things
14:05 up a little bit and I'm just gonna put a
14:07 few more dots . This is really all I want
14:09 to write down . So I'm writing down the series
14:12 in math speak . I don't want to put numbers
14:15 . I want to keep everything general . I have
14:16 a first term . The next term is just adding
14:18 one common difference . The next term is adding to
14:20 common differences and then they get all the way to
14:22 the end . I'm gonna have a final term and
14:24 then the term before that will be the final term
14:26 minus that common difference . And then there's terms all
14:28 in between . All right so far we haven't proved
14:30 anything but we just wrote a series down now again
14:33 I told you you can do anything you want in
14:35 math as long as it's legal and we're gonna we
14:37 want to try to recover that equation . So here's
14:39 how we're gonna proceed . We're gonna write the exact
14:41 same series down again underneath it . Except we're going
14:45 to reverse the order of all the terms . Because
14:47 if I add the terms up this way it's exactly
14:49 the same . Some as if I start here and
14:52 I add them in reverse . Just like two plus
14:54 three is five and then three plus two is five
14:57 . No matter how uh do the operation in the
15:01 edition , I'm gonna get exactly the same number .
15:04 So this some can be written a second way I'm
15:07 going to call it . I want to use a
15:08 different colour . I think I do this uh is
15:12 still S . N . I'm gonna get the same
15:14 answer but I'm gonna go in reverse so let me
15:16 start at the end and say the final term is
15:19 T . Seven . Okay what is the next term
15:22 going this direction ? It's this one , so it's
15:24 gonna be T . Seven in minus D . Like
15:28 this . And then if I were to continue going
15:30 backwards again what would be the term right before this
15:32 ? It would be T N minus two D .
15:35 Just like there's a plus here's a minus . So
15:37 it's gonna be uh T . C . N minus
15:40 two D . Like this . So starting from the
15:43 end it's minus D . And then minus D .
15:46 Again . And then I'm gonna put some dot dot
15:48 dots like this and then we're gonna go to the
15:51 other side . What is the end ? The end
15:53 point is T someone I'm gonna put t someone over
15:56 here what would be the term right to the right
15:59 of that guy ? Because I remember I'm writing my
16:01 son going this way . So the last term is
16:02 here the one right before that is this one .
16:04 So T . One plus deep you might say why
16:09 is he doing this ? I haven't explained why I'm
16:11 doing it yet . But what I'm telling you is
16:13 I can write the some of the terms as the
16:15 sum of all of these terms in the arithmetic series
16:18 . And then I can write the some of the
16:19 terms a second time , simply just reversing the order
16:22 uh in going and adding the terms in this direction
16:25 . So that the last term is T . One
16:27 and the first term is T . N . That's
16:29 all I've done here . And then what I'm gonna
16:31 do is I'm going to draw a line under all
16:33 of these right and I'm going to add them together
16:37 . How can I add them together ? Because remember
16:39 any equation with an equal sign ? If I have
16:40 two equations , I can add the left hand side
16:43 and I can add the right hand side together .
16:45 Just like a system of equations . You can solve
16:47 the system by addition add the left , add the
16:50 right . You may not understand why we're doing it
16:52 yet but you certainly know that it is legal to
16:54 do that because this is just an equation . And
16:57 so is this one ? Don't let the dots fool
16:59 you . Those are just terms in the middle .
17:00 It's perfectly fine to add these together . So what
17:04 am I gonna get ? S . N . Plus
17:06 S . N . Is going to be two times
17:08 S . N . Because I'm adding them together .
17:10 Just two of them . When I add these terms
17:13 together , what am I going to get ? I'm
17:15 gonna get T . One plus T . N adding
17:20 this term to this term . What's going to happen
17:22 when I add these together ? Okay . It's gonna
17:24 be T . One plus de plus T one dynasty
17:27 . But you see when I if I were to
17:28 add these together , the D . And the minus
17:29 T . Would disappear . That's why I actually flipped
17:32 it around to make the Ds disappear . So I'm
17:34 still gonna have these added together . I'm gonna have
17:36 T . One plus TN for the second term here
17:40 what happens if I add these together ? Same thing
17:43 the two D . And the negative two D .
17:44 Will go away but these will still be added together
17:46 so I will still have a T . One plus
17:49 T . N . Like this and then we skip
17:53 over to the end . What's going to happen here
17:54 ? The DSR gonna cancel . But I'm still gonna
17:56 have these I'm gonna have T . One plus T
17:59 . N . And then these guys are gonna be
18:01 added together . I'm gonna have T . It's T
18:03 . N . Plus one . I'm gonna write it
18:04 as T . One plus tion to match everything else
18:07 . So all I've done is added the left hand
18:09 side . And it turns out that when I add
18:11 the right hand side together all of the Ds cancel
18:15 in every single term is exactly the same . Every
18:18 term is the same . And how many of those
18:20 terms do I have ? I mean think about it
18:22 . I have 12345 all the way up to end
18:25 . There's in terms here . So when I add
18:28 them together , how many of these little parentheses am
18:30 I gonna have ? I'm gonna have in terms on
18:35 the right hand side and they're all the same .
18:37 So what does that mean ? That means on the
18:40 left hand side I'm going to have two times .
18:42 S Sir Ben on the right hand side , there's
18:44 any of these things so it's gonna be in times
18:47 what's inside T . One plus T . N .
18:51 These terms are identical but there's N . Of them
18:53 . So I'm multiplying by in finally I'm gonna solve
18:57 , I want to solve for the sum S .
18:59 Sub N . Is going to be in times T
19:02 . One plus T . N divided by two .
19:06 And if I remember correctly that's exactly what we have
19:10 right here . The some of the arithmetic series is
19:12 always the some of the first term in the last
19:14 term , divide by two , multiplied by end .
19:18 Now you might look at this and you might follow
19:20 this but you may say to yourself , I would
19:22 never , in a million years figured that out of
19:24 my own . Great , I'm glad you feel that
19:26 way . I wouldn't have figured it out of my
19:28 own either . I didn't wake up in the morning
19:29 one day and say I'm going to prove how arithmetic
19:32 series are added together . I didn't do that .
19:35 But what happens is people figure these things out and
19:37 they write them down and we learn from it .
19:39 So as long as you can understand it , that's
19:41 all I care about the fact that you may not
19:43 have known how to do it . That's okay .
19:44 There's tons of things , I don't know how to
19:45 prove off the top of my head , but I
19:47 can follow these steps and I can explain it to
19:49 you and you can understand it that this equation doesn't
19:52 just come from anywhere now . Really , the reason
19:54 this thing works , the reason you flip the thing
19:56 backwards is because there's a lot of symmetry in the
19:59 arithmetic series because you're adding a constant number . If
20:03 I flip the sequence of numbers around , then I'm
20:05 kind of subtracting and going backwards the other way .
20:07 Then when I add all that turns up , you
20:09 get a ton of cancellations , that is why it
20:11 works . Um and there's probably a million other dead
20:14 ends . You could go through to try to prove
20:15 this . That wouldn't be correct . But the bottom
20:18 line is , we know it's legal to write the
20:19 series this direction . We know it's legal to get
20:22 the same some by flipping the terms around . When
20:25 we adam we know it's illegal to add the left
20:27 and we know it's illegal to add the right and
20:29 we get all those cancellations which allows us to solve
20:32 for the sum . That's all we care about that
20:34 . We were able to prove that that is the
20:36 sum . All right . So now what we want
20:40 to do is we want to um solve a couple
20:45 of additional problems now that we have that basic idea
20:47 of what's going on . We want to find the
20:49 sum of these arithmetic series . So the first one
20:57 is in is equal to 20 . That means there's
20:59 20 terms . The first term is equal to five
21:04 . And the last term which is 20 because it's
21:07 20 terms is equal to 62 . And I ask
21:10 you give me the some of this . So all
21:12 I've told you is that I have the first term
21:14 in the last term . And how many terms I
21:15 have now ? I could write it go into a
21:18 calculator And I could write these terms out and I
21:21 could just add them up of course . But there's
21:22 20 of them . That's a lot of work .
21:24 I want to just find the number . I want
21:25 to figure out what it is . So you first
21:28 start out by writing down the equation . We just
21:29 proved the inthe sum of an arithmetic series is in
21:34 times T one plus TN first and last term's divide
21:37 by two . But I know in this case that
21:41 I'm looking for , the 20th partial sum . The
21:43 1st 20 terms here . And that means in his
21:46 20 And that means the first term is five .
21:51 The last term is 62 and I have to divide
21:54 by two because on the bottom here , that's what
21:56 the equation is always divided by two . So the
21:59 20th partial sum over here is going to be when
22:02 you take 20 and you add five plus 62 you
22:05 multiply by 20 and you divide by two , you're
22:07 going to get 670 . And this is the final
22:10 answer . Now we could stop there because that's really
22:13 all I asked you to do , but I think
22:15 it's good for us to right out what the terms
22:18 of this sequence would really look at , look like
22:21 . What are the terms of an arithmetic sequence ?
22:24 Right . The terms we learned a long time ago
22:26 , the in term is equal to the first term
22:29 , plus in -1 times the common difference . D
22:32 . Okay , so I've only given you the first
22:35 and the last term and I've given you how many
22:36 terms there are , but I haven't told you the
22:39 spacing of the terms . What's the common difference between
22:42 them ? And we never tell you that in the
22:43 problem statement because you don't need it to find the
22:45 sum . All you need to find the sum is
22:47 the first , the last term and the number of
22:49 terms . That's it . But to write out the
22:51 listing of terms , I need to know that spacing
22:53 the common difference . So let's put the final term
22:55 in to be 62 . Let's put the initial term
22:58 in 2 B5 . We know it's 20 terms away
23:01 . The first and the last term . So we'll
23:03 put in is equal to 20-1 . And then we're
23:06 gonna find the spacing . The common difference . If
23:09 we subtract this guy we're gonna get 57 is equal
23:13 to here . We have 19 times d . So
23:16 we take 57 divided by 19 , you get D
23:18 . Is equal to three . So the only way
23:20 this arithmetic series works , if you start at five
23:23 and 20 terms later you get 62 is if you
23:26 space out by three units , that means the sequence
23:28 will go like the series go like this . Five
23:31 uh Plus three more is eight plus three more is
23:34 11 plus three more is 14 plus dot dot dot
23:37 eventually you're going to arrive at 62 . And how
23:39 many terms are there ? If you write them all
23:41 out there's gonna be 20 terms of course you don't
23:45 need to write the series out in order to find
23:47 it some . That's the whole point . You want
23:49 to figure out what the sum is without doing all
23:51 that work . I'm just kind of showing you so
23:53 you kind of know and have in your mind exactly
23:55 what you're doing and so you can write the series
23:57 out if you ever need to . And the common
24:00 difference here is three plus three plus three plus three
24:03 and so on landing on 62 . All right .
24:08 All right , one more and then we'll call it
24:09 a day for this one . Mhm . What if
24:14 I give you a arithmetic series with 40 terms Where
24:18 the first term is negative 12 And the 40th term
24:24 Is equal 283 . This is a great example of
24:27 using this equation because I have 40 terms . If
24:30 I could write the series out and add them up
24:32 . But that's 40 editions . Certainly , it's going
24:35 to be easier to just apply the equation that we
24:38 know . So first thing is you write down the
24:40 equation , the inthe partial sum is equal to end
24:43 times the first term plus the last term divided by
24:46 two . Again , this is the average value of
24:48 the terms times the number of terms . And this
24:50 should not be uh to it should be T .
24:52 S event like this . Okay , So the 40th
24:56 some the 40th partial sum is going to be 40
24:59 times the first term which is negative 12 . The
25:03 last term here is going to be T sub 40
25:07 . Let me just double check myself . Yeah ,
25:08 t some 40 is 183 And we're divided by two
25:14 like this . So the 40th partial sum is going
25:17 to be four times what's in here . So 1
25:20 83 minus 12 times four you're gonna get 68 40
25:24 on the top and you'll divide by two . So
25:26 the 40th partial sum . The sum of all those
25:28 terms is 3420 . Notice I didn't even have to
25:33 know any of the terms in between . There's tons
25:35 of terms in between negative 12 and 183 . And
25:39 by the way , don't get so worried if some
25:41 of the terms are negative . So what some of
25:42 the terms start to the left hand side of zero
25:45 ? No big deal , but there's still space by
25:47 an even number or buy a whole by a constant
25:51 number . Let's go in just because we did it
25:54 for the last problem . Let's do that same thing
25:55 here . What is the equation to predict all the
25:58 terms of this series ? The term is the first
26:01 term for arithmetic plus uh , in -1 times the
26:06 common difference . Now , the last term we know
26:08 is 183 . The first time we know is -12
26:13 . We want to figure out We know how many
26:16 terms we have also 40 terms here , So it's
26:19 40 for N -1 . And we want to figure
26:21 out what the common spacing is here . So when
26:23 you take 183 and you add 12 to it ,
26:26 you're going to get 195 . And over here you're
26:29 gonna get 39 times D . And if you take
26:32 1 95 and you divide by 39 you're gonna get
26:35 a . D . Of five . That means these
26:38 terms are spaced out by five . You're never gonna
26:39 be able to figure this kind of thing out .
26:41 Just by staring at the numbers , you have to
26:43 use the equation to predict all the um uh Terms
26:47 in the sequence are in the series to figure that
26:49 out now that you know that the spacing is five
26:51 . You start at the first term negative 12 and
26:56 you add to it , The next term will be
26:58 five from there . So let's do it like this
27:01 negative 12 . The next term is going to be
27:03 negative seven because that's five more . The next term
27:06 is gonna be negative too because that's five more than
27:09 this . The next term after this will be positive
27:12 three because that's five more after this . And then
27:15 you're gonna have plus dot dot dot and you're gonna
27:17 land on the last term 183 . So if you
27:20 start here and add five and add five and add
27:23 five , add five all the way forward , do
27:25 it 40 times or 39 times really , Then you're
27:28 going to get to that 40th term which is 183
27:31 . This isn't really what I'm asking you to do
27:33 in the problem . I just like you in the
27:35 beginning not to just apply equations , you know ,
27:38 without knowing what's going on . I want you to
27:40 know that this is a sequence of numbers were adding
27:42 together and you only need to know the first and
27:45 the last term and the number of terms to tell
27:47 me what the sum is . But if you're ever
27:49 asked to write down the terms of the series ,
27:51 then you're going to need to be able to predict
27:53 and figure out the spacing so that you can write
27:56 those terms down . So in this lesson we've done
27:58 a lot of material , we already knew what a
28:01 series was in Math . We already knew that .
28:04 But now we want to start to add up and
28:06 want to find the sum . So we introduce the
28:08 concept of the partial sum when you're adding up to
28:11 terms three terms of four terms . The number underneath
28:13 just tells you how many terms are adding together when
28:16 you're adding up . In terms in other words ,
28:18 the equation to add up however many terms you want
28:20 is given by this , which we've discussed at great
28:23 length . And then we did the proof of it
28:24 and then we solve some problems . And so what
28:27 I want you to really pull out of this mostly
28:29 is how to apply this to real problems and also
28:32 to know that the proof of where it comes from
28:33 is not impossible to understand . All right . So
28:36 what I want you to do now solve these yourself
28:38 , go and tackle that proof if you want to
28:40 as well and then follow me on to the next
28:41 lesson will solve some more problems with the arithmetic series
00:0-1 .
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