04 -What is an Arithmetic Sequence? - Part 1 - Arithmetic Sequence Formula & Examples - By Math and Science
Transcript
00:00 | Hello . Welcome back . The title of this lesson | |
00:02 | is called arithmetic sequences . This is part one . | |
00:06 | Now . In the last few lessons we have introduced | |
00:08 | the concept of what a sequences Remember ? The sequence | |
00:10 | is just a listing of numbers . And we in | |
00:13 | general , introduced in the last couple of lessons , | |
00:15 | the arithmetic sequence in the geometric sequence . Mostly that | |
00:18 | was just to set the stage and let you know | |
00:21 | that there are two really , really important types of | |
00:23 | sequences , arithmetic and geometric so that you would have | |
00:25 | in one place the difference between the two . Now | |
00:28 | we're zooming in and focusing a little bit more in | |
00:30 | on the arithmetic sequence . So what I want to | |
00:33 | do in the beginning is right , the general equation | |
00:35 | down for an arithmetic sequence . And we're gonna spend | |
00:38 | some time understanding why it is the way it is | |
00:41 | . Because it's not always clear why the equation has | |
00:44 | written the way that it is . We want to | |
00:45 | make sure you understand that and then we'll solve some | |
00:47 | problems again , building your skills with arithmetic sequences . | |
00:50 | All right , so , let's just get to the | |
00:52 | punch line first in a textbook . Any textbook algebra | |
00:55 | calculus , uh geometry even sometimes covers this . You | |
00:59 | will come across the definition of the equation of an | |
01:02 | arithmetic sequence . So riff met IQ arithmetic sequence , | |
01:09 | right ? In general . And so what you will | |
01:10 | typically see for any for the uh in term of | |
01:15 | a arithmetic sequence is the following . The 10th term | |
01:19 | out in the future is equal to the first term | |
01:23 | of the sequence . Plus In -1 , multiplied by | |
01:27 | D . So that's it . Ladies and Gentlemen , | |
01:30 | you'll generally see this in a textbook somewhere . What | |
01:32 | we want to do is have you understand exactly why | |
01:36 | this equation Is the equation for the arithmetic sequence and | |
01:40 | why it makes total sense . It isn't always clear | |
01:43 | . For instance , why is it in -1 in | |
01:44 | there ? Why are we adding t one to the | |
01:47 | front there instead of doing something else ? Why is | |
01:50 | this thing represent an arithmetic sequence ? So remember the | |
01:53 | concept of an arithmetic sequence means it's a listing of | |
01:56 | numbers where each pair of terms each next door neighbor | |
02:00 | terms , we call them adjacent terms . They always | |
02:03 | differ by what we call it . Common difference , | |
02:05 | they always differ by some some constant number . So | |
02:08 | in the terms in the arithmetic sequence might differ by | |
02:11 | too , but that would mean every pair of terms | |
02:14 | next to each other has to be differing by two | |
02:17 | . Or they might differ by five or they might | |
02:19 | differ by seven . But whatever the number is , | |
02:21 | that difference between terms has to be the same difference | |
02:24 | for every term in the sequence . Otherwise it's not | |
02:27 | arithmetic . So we'll take a really simple example To | |
02:30 | illustrate what I'm saying here , if we have for | |
02:33 | instance , three and then seven and then 11 and | |
02:37 | then 15 . And ask ourselves what kind of sequence | |
02:39 | is this , then we can convince ourselves it is | |
02:42 | arithmetic . Why ? Because we look at pairs of | |
02:45 | terms , what is the difference between these terms you | |
02:47 | take ? The bigger term minus the term ? More | |
02:50 | this way minus this term . So seven minus three | |
02:53 | is four . So we say the difference between these | |
02:55 | terms , we call it D is equal to four | |
02:57 | units . But then when we subtract these terms , | |
03:00 | we again find that D is equal to four because | |
03:02 | 11 minus seven is four . And then for these | |
03:05 | two terms , we again find that D is equal | |
03:07 | to four because 15 -11 is four . So because | |
03:10 | the difference in these terms is the same as the | |
03:13 | difference in these terms is the same as the difference | |
03:15 | in these terms . This means it's an arithmetic sequence | |
03:18 | . Effectively , arithmetic sequences are just adding a known | |
03:22 | number to each term . So basically to get this | |
03:25 | term we add for then we add for then we | |
03:27 | add for that's what an arithmetic sequence is . All | |
03:30 | right . So just to formalize that we will say | |
03:33 | that this is arithmetic . Okay ? And d is | |
03:39 | equal to four . This is the thing we call | |
03:40 | the common difference . It means it's the difference between | |
03:46 | terms that is common to every pair of terms that | |
03:48 | you pick . Now . The question that I want | |
03:51 | to ask you is obviously this allows us to predict | |
03:55 | the next term . We could predict the next term | |
03:57 | pretty easily and then go on and on by adding | |
03:59 | for But what if I want to find the 100 | |
04:02 | term ? Obviously I could Continue adding for but I'm | |
04:06 | going to be adding forever . I'm gonna have to | |
04:08 | add for an ad for an ad for an ad | |
04:10 | for eventually I'll get to the 100 term and I'll | |
04:12 | have the answer that's boring . And it takes a | |
04:14 | long time . How do we write a formula down | |
04:17 | to predict what the 100 term will be without sitting | |
04:20 | there and adding and adding constantly over and over again | |
04:23 | . So let's talk about fine . How to find | |
04:28 | T sub 100 because this is T someone T sub | |
04:32 | two T sub three TC . Before these are the | |
04:34 | different terms . How to find the 100 term . | |
04:37 | How do we do it ? All right . So | |
04:40 | let's kind of write this down . We have three | |
04:42 | . Then we have seven . We have 11 . | |
04:45 | Then we have 15 . Right ? And we have | |
04:48 | a bunch of terms in between . That's what the | |
04:50 | dots mean . Some unknown number of terms . Well | |
04:52 | , it's not unknown , but a bunch of terms | |
04:54 | in there . Finally We have T sub 100 . | |
04:58 | So overall there's 100 terms . There's 100 terms in | |
05:04 | the sequence . Right ? This is T one T | |
05:07 | two , T three , T four . That we | |
05:08 | have 56789 10 . So on all the way to | |
05:11 | T sub 100 . And we just said that the | |
05:15 | difference between this is four and the difference between this | |
05:19 | is four , and the difference between this is four | |
05:21 | . And because it's arithmetic , every term in the | |
05:24 | sequence differs by four . All the way up to | |
05:26 | the 100 term , that's what we're saying . But | |
05:28 | let me ask you a question . Obviously this is | |
05:30 | termed 123 and four . And we have a difference | |
05:33 | here , in a difference here , in a difference | |
05:35 | here , how many differences between terms are we going | |
05:38 | to have all the way out to the 100 term | |
05:41 | ? And I think you can convince yourself that basically | |
05:44 | between this term and this term , whatever the next | |
05:47 | one after 15 is , is the difference of four | |
05:49 | and then right before the 100 term is the 99th | |
05:52 | term , the difference there is also for so if | |
05:55 | I look at how many differences I'm going to have | |
05:57 | all together , I'll draw a big little curly brace | |
05:59 | right here . There are 99 differences . It takes | |
06:07 | a second for you to realize that . But basically | |
06:08 | , if you have 100 numbers And you're only subtracting | |
06:12 | pairs of numbers , then how many subtractions do you | |
06:15 | have ? Because you're taking them two at a time | |
06:17 | like that ? There's not 100 differences . There's 99 | |
06:21 | differences . You see , because you can see the | |
06:23 | way it is here , because you're taking pairs , | |
06:25 | that's 1234 And then keep going all the way here | |
06:28 | . There's not gonna be 100 there'll be 99 differences | |
06:31 | right there . So if I wanted to just figure | |
06:34 | out what the 100 term is , If I wanted | |
06:38 | to figure out what the 100 term is , how | |
06:39 | would I do it ? If I wanted to do | |
06:41 | it manually I would start with the first term and | |
06:44 | I would add four and add four and add four | |
06:46 | and four . And before I could add literally I | |
06:48 | could put plus four plus four plus four . How | |
06:50 | many times would I do it ? I would do | |
06:51 | it 99 times . I would have 99 fours Here | |
06:54 | . We have a shortcut for that . It's called | |
06:56 | multiplication . So I can say I'm gonna say 99 | |
07:01 | Times four . So what I'm doing is I'm starting | |
07:04 | with the first term and I'm adding to it 99 | |
07:07 | quantities of four , right ? And that is going | |
07:10 | to be the value of whatever t 100 is the | |
07:14 | 100 term is . I start at the beginning and | |
07:16 | I add for 99 times . And that's gonna give | |
07:19 | me this guy here , you see , this guy | |
07:21 | is exactly the same form as what I have over | |
07:23 | here . So I can generalize That . If I | |
07:28 | don't want the 100 term , if I just want | |
07:31 | the 10th term , whatever that number is , then | |
07:33 | instead of the number three here , I'll just start | |
07:35 | at whatever the first term in the sequences and I | |
07:38 | will add to that . What am I going to | |
07:40 | add to that Notice this was 99 , I had | |
07:43 | 99 differences , I had 100 terms and 99 differences | |
07:47 | . That means that the number in here is how | |
07:49 | whatever term you're trying to reach -1 , that's how | |
07:52 | many differences are there ? And you're gonna multiply that | |
07:55 | by the common difference d . So this is the | |
07:59 | exact thing that we wrote down for the definition of | |
08:01 | the arithmetic sequence . But whenever I show it to | |
08:03 | you , just cold like that it doesn't make a | |
08:06 | lot of sense . Why is it in minus one | |
08:08 | ? Why are you multiplying by D . And why | |
08:10 | do you add T sub one up in the top | |
08:12 | like this ? The reason why is because what you're | |
08:15 | doing to find the 10th term is you just start | |
08:18 | at the first term and add to that . Those | |
08:22 | differences . How many differences ? Because that's the difference | |
08:25 | between the terms . How many differences do you add | |
08:27 | ? You add in -1 of them . So if | |
08:29 | you're trying to reach the 1000 term then you have | |
08:33 | 999 differences times whatever the differences . And that's what | |
08:37 | you're adding to the first term . And this is | |
08:40 | why this thing is called the definition of the arithmetic | |
08:44 | sequence . I want you to burn this in your | |
08:46 | mind so that whenever you look at this equation you | |
08:49 | don't just memorize the equation . And so I'm going | |
08:51 | to use it . I want you to look at | |
08:53 | it and say this is what it's telling me . | |
08:55 | Start at the first term , add differences to it | |
08:57 | , How many do I add ? I add ? | |
08:59 | We'll have returned , I'm trying to get to minus | |
09:00 | one because that's how many differences I have there . | |
09:03 | That's what it's doing . It's just doing all of | |
09:05 | the work of getting all the way to that term | |
09:07 | without you have to add over and over and over | |
09:09 | because you just multiply to get there . Okay so | |
09:13 | that's the background as to why this is the arithmetic | |
09:15 | sequence . Now , what we want to do is | |
09:17 | solve a couple of problems . So what I'm gonna | |
09:19 | do is write a sequence down on the board and | |
09:21 | I'm going to ask you to find the term of | |
09:23 | the sequence . If you ever have a problem that | |
09:26 | it says give me the term of the sequence , | |
09:28 | it doesn't mean that they want a number , it | |
09:30 | doesn't mean that I want you to tell me Term | |
09:32 | number 10 . If I tell you find the term | |
09:35 | I want an equation that will give me any term | |
09:39 | I want if I put the number . NN so | |
09:41 | that it gives me an equation that's general for whatever | |
09:43 | term I'm trying to get to . So for instance | |
09:46 | , if I come over here and say here's a | |
09:49 | sequence 24 , 32 40 and 48 and it goes | |
09:56 | on and on from here , what I want you | |
09:58 | to do is give me an equation or a the | |
10:01 | term of the sequence . That would allow me to | |
10:03 | predict any term that I want down the road . | |
10:06 | Give me an equation that I could for instance pick | |
10:08 | the or calculate the one million term . So you | |
10:11 | need to find uh you need to use this arithmetic | |
10:14 | definition here , the arithmetic sequence definition . You know | |
10:19 | or you suspect this is gonna be arithmetic , so | |
10:22 | you need to find that common difference . That's the | |
10:23 | first thing . 30 to minus 24 what is that | |
10:27 | equal to ? That's going to be eight . Right | |
10:30 | . Uh what is 40 -32 ? That's again eight | |
10:33 | . What is 48 -40 ? That's again eight . | |
10:35 | So it's obviously an arithmetic sequence . The common difference | |
10:38 | is eight . So what you need to do is | |
10:40 | write down that the first term t someone is 24 | |
10:46 | . And the common difference common to all of these | |
10:49 | terms here is eight and then we go back and | |
10:52 | use the equation for the arithmetic sequence but hopefully it's | |
10:56 | not just an equation , it makes sense to you | |
10:58 | . The 10th term down the road is just gonna | |
11:00 | have me starting at term number one and adding to | |
11:02 | it a bunch of differences . How many of these | |
11:05 | differences do I need ? Well I need in minus | |
11:07 | one of them times . Uh The common difference D | |
11:11 | right so I'm just starting here and I'm adding so | |
11:13 | many of these differences now for our particular example the | |
11:16 | first term was 24 And then in is uh is | |
11:22 | just in you leave that alone in -1 times the | |
11:25 | common difference which is eight . Okay , so then | |
11:28 | you say you have to just do the algebra to | |
11:30 | simplify everything , 24 plus this eight gets distributed in | |
11:34 | and becomes eight in and then eight times negative one | |
11:37 | gives you minus eight . So T seven is you | |
11:39 | have a number in a number I can uh subtract | |
11:42 | or add however you want to look at that together | |
11:44 | . So what I'm going to have is eight in | |
11:46 | minus 16 Actually a 10 plus 16 . So 24 | |
11:52 | -8 gives me the 16 . This stays along for | |
11:53 | the ride , this is the equation for the term | |
11:57 | of this arithmetic sequence . And you kind of need | |
12:00 | to get used to seeing that something like this , | |
12:03 | something times in plus something in general is going to | |
12:05 | be or or it's going to create an arithmetic sequence | |
12:09 | . Now it's always a good idea when you write | |
12:11 | the term of the sequence down to make sure it | |
12:13 | works . So the easiest way to do that is | |
12:15 | to put in his equal one . What happens if | |
12:17 | n is equal to one ? Eight times one is | |
12:19 | eight and 16 plus eight you get 24 . So | |
12:21 | we say t someone from this calculation Should be 24 | |
12:26 | . That's exactly what T someone is . If we | |
12:30 | put these up to in here eight times two is | |
12:32 | 16 plus 16 is 32 which is exactly what this | |
12:36 | is . And if you put a piece of three | |
12:38 | and tisa four you'll get these . And this also | |
12:40 | allows you to calculate what T sub 95 is or | |
12:43 | T sub 1000 and 30 for whatever you want . | |
12:46 | You don't have to keep adding and adding and adding | |
12:48 | . You just use the equation of the arithmetic sequence | |
12:50 | and this is why it's doing that . You're starting | |
12:52 | at the first term and you're adding to it . | |
12:54 | However many differences you need to get there . All | |
12:57 | right , So all of the remaining problems in this | |
13:00 | lesson are all going to be similar . I'm just | |
13:03 | gonna give you practice what if the sequence is negative | |
13:05 | three , negative 10 , -17 . Negative 24 dot | |
13:12 | dot dot . Give me an equation for the In | |
13:14 | terms of this sequence first , see what the common | |
13:17 | differences , negative 10 minus a negative three . The | |
13:22 | difference here is gonna be negative seven . Another way | |
13:25 | to look at that is just a negative three plus | |
13:28 | . Something is going to give me the negative 10 | |
13:29 | . It has to be negative seven . The common | |
13:32 | difference here , negative 17 minus the negative 10 becomes | |
13:35 | plus and so again the common difference will be seven | |
13:37 | , negative seven and then same thing here negative 24 | |
13:40 | minus the negative 17 becomes plus 17 . So what | |
13:43 | you get again this negative seven . Again , you | |
13:45 | should be able to start here and add a negative | |
13:47 | seven to get this . Take this add a negative | |
13:49 | seven to get this . Take this add a negative | |
13:52 | seven to get this . So the common difference D | |
13:54 | . Is negative seven . Okay In term number one | |
14:00 | is negative three . Common difference is negative seven . | |
14:03 | And then the arithmetic sequence at the 10th term is | |
14:07 | just whatever the first term is plus In -1 times | |
14:11 | the common difference d . So then I just put | |
14:15 | it in what I know the end of term is | |
14:19 | the first term negative three plus in minus one times | |
14:23 | D . Negative seven . Now you have to be | |
14:25 | careful with the signs but not a big deal negative | |
14:28 | seven times in is negative seven in negative seven times | |
14:32 | negative one is positive seven t seven . Now you | |
14:35 | have negative three positive seven that's gonna give you four | |
14:39 | -7 . usually I like to flip it around so | |
14:43 | make its negative seven . N plus four . You | |
14:45 | could leave it like this if you want . Generally | |
14:47 | I would like to write the end first but there's | |
14:49 | really no answer . I'll just keep it at negative | |
14:51 | seven plus four . You could of course write it | |
14:52 | like this . If you want to always a good | |
14:55 | idea to check what would t someone be . If | |
14:59 | I put them one in here then that's going to | |
15:01 | give you negative seven . Negative seven plus four is | |
15:04 | negative three . That's exactly what the first term is | |
15:07 | . What is T . Sub two . Then if | |
15:09 | I put two I'll get negative 14 but negative 14 | |
15:12 | plus four is gonna give me negative 10 . That's | |
15:15 | exactly what term number two is . And I'll let | |
15:16 | you go down term three in turn four you'll recover | |
15:19 | all of these other terms here . It's always a | |
15:21 | good idea , as I've said several times , to | |
15:23 | double check that the sequence equation that you have come | |
15:27 | up with actually does generate the terms of your sequence | |
15:31 | . All right , one more to kind of get | |
15:35 | warmed up here . Yeah . What if the sequence | |
15:38 | is five , eight , 11 , 14 . And | |
15:43 | then on and on . Like this give me an | |
15:45 | equation for the term of the sequence . Well what | |
15:49 | is the difference here ? Five minus ? I'm sorry | |
15:51 | eight minus five is three , This minus this is | |
15:55 | three And this minus this is again three . So | |
15:58 | you know the common differences three and you know the | |
16:00 | first term is five . Then it goes back to | |
16:05 | straight into the arithmetic sequence formula . The Inthe term | |
16:09 | is the first term plus in -1 times . Deep | |
16:14 | in term is the first term which is five plus | |
16:19 | In -1 times . So five goes here , three | |
16:22 | goes there and then multiply through . We have 53 | |
16:27 | times in is three in three times negative one is | |
16:30 | negative three . And then you add your numbers together | |
16:32 | . I'm going to write the three in first five | |
16:35 | minus three is positive two . So you get three | |
16:38 | in plus two and this is the correct answer . | |
16:41 | And it's always a good idea to double check yourself | |
16:43 | . What would t someone be put a one here | |
16:46 | three plus two gives you five . That's what I | |
16:49 | have here . T sub to uh put it to | |
16:52 | in here . Five plus two is seven . Uh | |
16:55 | I put it to in here six plus two . | |
16:57 | Of course it's not seven and eight . So that's | |
17:00 | correct . So put it to in here two times | |
17:01 | three is six plus two is eight so it recovers | |
17:04 | here and if you put three in you have nine | |
17:06 | plus two is 11 and so on . You can | |
17:07 | recover the rest of the terms . Are any of | |
17:10 | these less any of these problems ? Rocket science . | |
17:12 | No they're not rocket science . But what usually happens | |
17:15 | is when a student solving problems they can kind of | |
17:19 | do the kind of more basic ones like what we're | |
17:21 | doing here , but when we get to anything a | |
17:23 | little more complex , their knowledge completely breaks down because | |
17:27 | they never practiced the fundamentals and they didn't really understand | |
17:30 | what this arithmetic sequence formula really was doing . So | |
17:33 | what I want you to do now is solve all | |
17:35 | of these problems yourself . Make sure you can solve | |
17:37 | them all . You get all the correct answers . | |
17:38 | Follow me on to the next lesson . We will | |
17:40 | increase the complexity a little bit to show you how | |
17:42 | to solve arithmetic sequence problems of slightly higher complexity . |
Summarizer
DESCRIPTION:
Quality Math And Science Videos that feature step-by-step example problems!
OVERVIEW:
04 -What is an Arithmetic Sequence? - Part 1 - Arithmetic Sequence Formula & Examples is a free educational video by Math and Science.
This page not only allows students and teachers view 04 -What is an Arithmetic Sequence? - Part 1 - Arithmetic Sequence Formula & Examples videos but also find engaging Sample Questions, Apps, Pins, Worksheets, Books related to the following topics.