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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