U14L2+Arithmetic+vs.+Geometric May 2.mp4
Okay arithmetic versus geometric so one of them you're adding something each time one of them you're multiplying something each time i'd like you to theorize which one of them feels like it should be adding you gotta pick a team.
How many of you are feeling arithmetic feels like adding you were right that's the one that's adding arithmetic arithmetic adding geometric multiplying can you see how this is multiplying by two each time that's a geometric sequence the others.
Is adding two each time that's arithmetic and i know it's been a few days you know you've been kind of out of the game so let's see if you remember u one equals five that means we start with five u n equals u n minus one and then if i said plus seven do you get that would mean i add seven each time.
If you understand this you should be able to tell me the first three terms would you please find the first three terms it is as simple as you might think jp what's first term hint it's given and then the next term use your fingers and toes if you want to.
No we add seven 12 and then add seven again there you go those are the first three terms raise your hand if you had that okay good then let's do a that's arithmetic because you're adding let's do a geometric u one equals ten u sub n equals you know what always comes next always as in every single.
Time u n minus 1. what it's saying is the term right in front of this term the term right in front of this term times three but we put a parenthesis when we're going to times something would you tell me what the first three terms are of this.
J james what's the first term of it yup and then how do i get the second term that's times by three so you just times by three and that's 30. kennedy how does it finish yep that makes sense to you guys.
Those are called recursive and that's a recursive sequence this sequence will go on and on and on the first one is called u sub one the second one's called u sub two etc if you pick a random one out of a let's say that later on it was one 120 and 240.
And 360. i'm adding 120 each time by the way looked like i might have been timesing by two at the beginning but i was actually adding 120 each time is this arithmetic or geometric this is the kind where you add so what's it called arithmetic.
And i know i'm you're thinking the word arithmetic that's back in elementary school when you're doing this unit it's called arithmetic and i know it feels wrong to say a word differently than you've always said it but i know it's spelled the same as arithmetic but you say arithmetic this is an arithmetic sequence would you.
Please write the equation it starts with u1 and pause for a second okay so we're going to start with u1 is 120. and then this is the next line you always write this u sub n which is like any random term like say i picked this one u sub n that i picked that one.
Pick any term you want it's given by u sub n minus 1 which is like the term right in front of it except you gotta add 120 so we put plus 120. does that make sense to you guys then i introduced on the last day we had class right before the weekend i introduced this concept of.
N which is the little sub numbers you know like this like this like this and has to be greater than or equal to two that's called a domain you will have to write one of these on your equations.
Otherwise you'll lose a minimum of five percent off if you don't include that and i know i haven't been stressing that and i know most you didn't put it but did you please go put it right now put it in with your equation that you just wrote because without the domain without this.
You won't get credit it's only it's it's an integral part of the problem what it's saying is that it only works starting on n is 2. think about that n is equal to 2 is the smallest it can be which means this will not predict the first term the first term how can you get the first term do you go to term in front of that.
No there isn't one so recursive sequences have a hard start and they do go on forever but you can't like make up a number that existed before this so this only works to predict the second term and the third term in the fourth term so use it's saying n has to be at least two or this equation.
Won't work and if that was too complicated for you just always say n is greater or equal to two it's all you gotta write and it means the second term is where this thing starts working so if i said like i'm not telling you the first term.
When you can't do these knowing that you add 5 each time doesn't help unless you know the first term and will our formula predict our first term no so in other words if i had just this part can you tell what the first term is from that.
No you can't see this formula does not tell you the first term that's why you always got to say at the end n has to be greater than equal to 2. n are the sub numbers and n sub 1 we don't know u sub 1 has to be given you have to tell us oh yeah start with a 7 and then we'll add 8 each time 7 plus 8 15 plus 8.
23 dot dot dot but you can't tell what the first term would be that's what this is saying i almost wish they said n cannot equal one because then it would be like this won't work for the first term.
But it's kind of like that saying n has to be two or bigger arithmetic or a special type of sequence where each term is equal to the previous term plus a constant you know that number we keep adding it's called the constant and it's called the const common difference i almost wish they called it.
A constant difference but the common difference c4 is the common difference here you might be like that's not a difference that's an ad problem you're right differences or subtract problems so you go nine minus five and it gives you four.
Five minus one gives you four difference subtract 17 minus 13 it gives you four so this is the difference difference means subtract and the common difference was four what's common difference here well i know it's subtracting 2 each time.
Is 18 minus 20 negative 2 look at that the common difference again the difference means subtract 16 minus 18 it's negative 2. so negative 2 is the common difference whereas if you're doing geometric kind where you're timesing by 3 then this divided by this 6 divided by 2.
Equals 3 is the common ratio did you know that fraction when you two things divided that means a ratio it's pretty much the same thing as a ratio so the ratio that they all have in common is that if you take this divide by this you get 3. this divided by this you get 3. 162 divided by 54. i'll trust them i don't really know but if i used a.
Calculator ipad it would be three the common ratio seems to be three oh hey here's the ultimate test do you understand these words if you do you can write an equation where you go u sub 1 equals see if you can write the equation for this i'll pause for a second.
It's a typical test question here we go u sub 1 is negative 12 good job keep going common difference is plus five plus five and then what's that little domain thing that i said is gonna be worth at least five percent on the test and greater than or equal to two yes that's gonna be really annoying and.
You're gonna forget it and if you do you're gonna lose five percent at least so please double check for that at the end yes yep just write it off to the side it's kind of like in more complicated math formulas where you go like this they'll often say something like oh by the way a cannot be zero because do you.
Get you're not supposed to divide by zero and so they'll say this like hey you can't do that by the way so same thing it's like hey this only works for n is two or bigger let's not always say that u sub 1 it can't know that because that's the starting term and just because i say.
Like hey you're going to add 5 each term that doesn't tell you where to start so this formula only works for n is greater than equal to okay write a recursive formula where the first term is 20 and the common ratio different that's a different type of question.
Still use of one equals still u sub n still u sub n minus 1. you know what that's in every single formula and i can add something to that every single formula always says n is greater than or equal to 2 on it now all you got to do is figure out what goes here in the red question mark and what goes here in the blue question.
Mark that's really it so connor what's the first thing the red question mark what is our first term nope read it closer what's the first term 20. and the point four.
Be careful with that now should i add point four subtract point four multiply by point four or divide by point four you put it in parentheses i'll give you that hint does that make you think add subtract multiply or divide yep that's a multiply.
There's your answer do you get that the black part of the answer would have always been there no matter what the problem had said they always have these pieces in a contract if you were a lawyer they would say that's the boilerplate it's like it's in every contract every contract always has to say who the two.
Parties are that are going into the contract if it's me and tesla because i'm buying a tesla then the contract will be between me and tesla and then i agree to do a certain thing and they agree to do a certain thing like i might agree i'm going to spend this much money i'm going to give it to.
You and tesla might agree in exchange we are going to give you this car and this software and agree to support that car until it's death you know because if they just gave me the car and said yeah we're not going to update the software i'd be pretty bad because then you know car doesn't work one day and all of a.
Sudden i can't fix it so so i'm sure in the contract it says that they'll give me the car and support the software for as long as the car maybe they even have a year on that like we'll support the software for 20 years or something like that okay.
So in a formula the boilerplate is this will always be there this will always be there and this will always be there do you get that really only means you have to figure out two things i think you got to figure out where does it start and what do you do each time and this is a common ratio question which told me.
It was a geometric so here's the words you got to know common ratio goes with geometric common difference goes with arithmetic alexis could you shut that door it's getting a little loud for lunch time out there okay now.
This is something there's an exception to every rule u1 versus u0 i just got done saying that u1 is how we're always going to start and it's kind of like other things in life where you say would you agree if you're telling a first-time driver you would say.
You always want to keep the car between the lines on the highway right first time driver you're going to say that but isn't there always exceptions like well if there's a car coming at you head-on you better go across those lines and get.
Out of the way right or if there's a person standing in the road you're not just gonna hit them and say well you told me to keep it between the lines i couldn't go across the line so there's always an exception my point we usually start with u sub 1 but we can start with u sub zero.
If i said u sub 0 equals 5 and u sub n equals u sub n minus 1 plus 6. what would be different well it's kind of the same thing i still start with five it's just that that's called u sub zero and when i do add six and get eleven that'll be called u sub one and isn't a.
Little weird to start with zero instead of one yes so when would you ever do this would you agree that an experiment that you start right before the experiment starts that's at time zero we can't call the start of an experiment.
One second there's a gap between zero seconds and one second it's actually a fairly long time in photography do you think that the aperture on your camera stays open for a whole second no it gets over with in less than half of a second like probably less than a tenth of a.
Second it just goes super fast but it takes some time my point is that if you're starting an experiment you start it at time zero and after one second maybe the temperature's already gone up because you're heating this thing and it starts at whatever temperature but then after one second it's already changed so sometimes we want to start with u sub.
Zero when there's an experiment so if we were gonna do an experiment and say i'm gonna keep a heater on this uh vial of liquid and you're going to keep heating it until it gets to like 400 degrees you're going to start at some point and after one second has gone by they'll be.
It'll be hotter than it was before so we have to start at time zero so that's why you sometimes start with u sub zero and all that means is that the first term then isn't called u sub one it's called u sub zero that's our first term this is the second term even though it's.
U sub one it's still a second term i know it kind of freaks you out a little bit but this is called the first term whether or not you start with u sub zero the problem will tell you or.
You'll know because it's an experiment or a real world scenario all right looks like a real world scenario a concert hall has 59 seats in row one so imagine for a moment this is our concert hall there's a whole bunch of seats here and i'm not going to be able to draw out.
Enough x's to make it actually 59 but there's 59 in this row and in the next row a whole bunch of seats it's going to be a little bigger if you've ever seen a concert hall it can get bigger like this or it can anyways this one's 63. this one's 67.
How are you noticing that it's like a sequence where we start with 59 it goes to 63 it goes to 67 what's happening each time between 59 and 63 how much is that adding four how much does it add this time then that's what's called a common difference.
It sounds like something i could write an equation for doesn't it in fact let me just do that it'd be u sub i'm going to choose u sub 1 for the first row that feels like it makes sense to me row one otherwise how the heck am i going to do row zero that'd be weird because i'm going to say you u1 is 59.
Un equals u n minus 1 and then i add 4 each time and if they ask me like hey how many are there in the 10th row i could say it was 59 first and then i add 4 is 63 and then i add 4 is 67 and then i can just keep doing that until i found the 10th one i showed you how to.
Use a calculator on that right you go 59 plus 4 enter plus 4 enter and now your calculator knows oh he's done it twice in a row i think he's going to keep adding four and just go enter enter enter enter until you finally get the tenth one.
But you've probably been suspecting that there's a faster way to do that you will see all right so here's our recursive formula is this arithmetic or geometric say it come on arithmetic yep because you're adding four each time.
How many seats are in the fifth row well that's three rows so i can just go two more now let me ask you what if you had to do this and find the 25th row i'd use a calculator now is there a faster way than with a calculator there is but it involves.
Something kind of complicated and i'm not sure which way we're supposed to like go with yet so i don't want to launch it and tell you some complicated way and then there's two ways you can do it i don't know which way we're gonna use for our test so give me time eventually i'll show you.
The fast way but if i just keep adding 4 here 71 75 there's 75 seats in that row okay i'm gonna give you this hint you usually start with u sub zero or u sub one you always have u n equals u n minus one and then if you're multiplying by.
Something it goes here or if you're adding by something it goes there too but it doesn't have parentheses read this scenario and see if you get it i bet you'll figure out what to start with i'll bet you'll figure out whether you multiply or add i bet you can do it because there's really only two things left the blue.
Question mark and the black question mark i'll pause for a second i want to see if you can do it all right so let's see arithmetic or geometric well it looks like they were going increasing by two percent i call that multiplying by 1.02 did you already forget last unit.
That would have been times by 1.02 starting at 79.99 how many of you figured that out raise your hand if you did okay and i bet some of you knew it was 79.99 but you weren't sure exactly how to do the two percent thing does that make sense to you now.
Is it arithmetic or geometric well you're timesing by something so it's geometric and there's our formula oh i keep saying this you no no not you what's the domain thing you gotta have at the end or you lose five percent and is.
It's easy to forget isn't it but did anybody have that anybody in this whole room nice three people cool all right careful with geometric increase means the numbers need to get bigger so that the common ratio needs to be bigger than one see what a lot of.
People put is 0.02 in there if you actually times by 0.02 it's going to shrink a lot so you need 1.02 not just 0.02 that's what they're trying to say there common mistake is to say just 0.02 wrong 1.02 even better 1 plus 0.02 but it's the.
Same thing okay i feel like you understand it i checked your homework there's only one problem that i feel like would be logical to chop off at the end chop off the last question find your homework if you haven't already i know some of you started early.
Which isn't a crime as long as you're still paying attention kennedy can you read me the first problem did you catch what they were doing each time and is that a common ratio or a common difference yep.
And what do they actually ask you about this problem oh okay so arithmetic is an a or a g what do you say i agree and what is the common difference i'm going to call it the cd yep and you can say plus or you can just say.
Five as long as you didn't mean minus so like positive five we usually can just say five is there any other parts is just that it's just saying the value of common different ratio again okay and that's the five all right okay.
I think you're getting the idea when would you ever use u sub zero if you're doing an experiment or in some real world problems if you start with u sub zero it just makes it easier to understand that like when we started our experiment the tesla bot was able to last for four hours before.
Charging but after one hour he had three hours of charge left started with four hours of charge after one hour he has three hours of charge you see how then starting with u sub zero would make sense if you already do.
An experiment like that okay all right that's all i have for you for today