04 -What is an Arithmetic Sequence? - Part 1 - Arithmetic Sequence Formula & Examples - Free Educational videos for Students in K-12

04 -What is an Arithmetic Sequence? - Part 1 - Arithmetic Sequence Formula & Examples - By Math and Science

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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 .

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