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Current time:0:00Total duration:8:18

Video transcript

what I want to do in this video is familiarize ourselves with the notion of a sequence and all sequences is an ordered list of numbers so for example I could have a finite sequence that means I don't have an infinite number of numbers in it or let's say I start at one and I keep adding 3 so 1 plus 3 is 4 4 plus 3 7 7 plus 3 is 10 and let's say I only have these 4 terms right over here so this one we would call a finite finite sequence I could also have an infinite sequence so an example of an infinite sequence let's say we start at 3 and we keep adding 4 so go to 3 to 7 to 11 15 and you don't always have to add the same thing we'll explore fancier sequences 1 the sequence is where you keep adding the same amount we call these arithmetic sequences which we will also explore in more detail but to show that this is infinite to show that we keep this pattern going on and on and on I'll put three dots this just means we're going to keep going on and on and on so we could call this an infinite infinite sequence now there's a bunch of different notations that seem fancy for denoting sequences but this is all they refer to but I want to make us comfortable with how we can denote sequences and also how we can define them so we could say we could say that this right over here is the sequence a sub K a sub K for K is going from 1 to 4 is equal to this right over here so when we look at it this way we can look at each of these as the terms in the sequence and this right over here would be the first term we would call that a sub 1 this right over here would be the second term we call it a sub 2 I think you get the picture a sub 3 this right over here is a sub 4 so this just says all of the a sub KS from K equals 1 from our first term all the way to the fourth term now I could also define it by not putting explicitly writing the sequence like this I could essentially do it using defining our sequence as explicitly using a kind of a function notation or something close to function notation so the same exact the same exact sequence I could define it as a sub K from K equals 1 to 4 with with instead of explicitly writing the numbers here I could say a sub K is equal to some function of K so let's see what happens when K is 1 we get 1 when K is 2 we get 4 when K is 3 we get 7 so let's see when K is 3 we added 3 twice let me make it clear so this was plus 3 this right over here was a plus 3 this right over here is a plus 3 so whatever K is we added we started at 1 and we added 3 1 less than the K term times so we could say that this is going to be equal to 1 1 plus 1 plus K minus 1 1 plus K minus 1 times 3 or maybe Baxter right 3 times K minus 1 same thing 3 times K minus 1 and you can verify that this works so you can verify that this works if K is equal to if K is equal to 1 you're going to get 1 minus 1 is 0 and so a sub 1 is going to be 1 if K is equal to 2 you can have 1 plus 3 which is 4 if K is equal to 3 you get 3 times 2 plus 1 is 7 so it works out so this is what this is one way to explicitly define our sequence with kind of this function notation I want to make clear I've essentially defined a function here if I wanted in more traditional function notation I could have written a of K where K is the term that I care about a of K is equal to 1 plus 3 times K minus 1 this is essentially a function where an allowable input the domain is restricted to two positive integers now how would I denote this business right over here well I could say that this is equal to this is equal to and people tend to use a but you could I could use the notation B sub K or anything else but I'll do a again a sub K and here we're going from K from our first term so this is a sub 1 one this is a sub to all the way to infinity or we can define if we wanted to define it explicitly as a function we could write this sequence as a sub K where K starts at the first term and goes to infinity with with a sub K is equally so we're starting at three we're starting at three and we are adding four we're adding four one less time for the second term we added four once for the third term we add four twice for the fourth term we add four three times so we're adding four one less than the term that we're at so it's going to be plus four times four times K minus one four times K minus one so this is another way of defining this infinite sequence now in both of these cases I defined it as an explicit function so this right over here is explicit let me that's not enough that's not an attractive color let me write this in these are this is an explicit function and so you might say well what's what's another way of defining these functions well we can also define it especially something like this Erath we could all like an arithmetic sequence we can also define it recursively and I want to be clear not every sequence can be defined as either an explicit function like this or as a recursive function but many can including this which is an arithmetic sequence where we keep adding the same quantity over and over again so how would we do that well we could also another way of defining this first sequence we could say a sub K starting at K equals 1 and going to 4 with and when you define things recursively when you define a sequence recursively you want to define what your first term is with a sub 1 a sub 1 equaling 1 and then you can define you can find every other term in terms of the term before it and so that we could write a a of K plus or let me write it this way a sub K a sub K is equal to the previous term is equal to the previous term so this is a sub K minus 1 so a given term is equal to the pre via Sturm let me make it clear this is the previous the previous term plus the previous term plus in this case we're adding three every time we're adding three every time now how does this make sense well we're defining what a sub one is and if someone says well what happens when K equals two well they're saying well it's going to be a sub two minus one so it's going to be a sub one plus three well we know a sub 1 is 1 so it's going to be one plus three which is four well what about a sub 3 what's going to be a sub 2 plus 3 a sub 2 we just calculated is 4 you add 3 is going to be 7 this is essentially what we mentally did when I first wrote out the sequence when I said hey I'm just going to start with 1 I'm just going to start with 1 and I'm just going to add 3 for every successive term so how would we do this one well once again we could write this as a sub K starting at K the first term going to infinity with our first term a sub 1 is going to be 3 now and every successive term a sub K is going to be the previous term the previous term a sub K minus 1 plus 4 plus 4 and once again you start at 3 and then if you want the second term it's going to be the first term plus 4 it's going to be 3 plus 4 you get to 7 and you keep adding 4 so both of these this right over here is a recursive definition recursive we started with kind of a base case and then every term is defined in terms of the term before it or in terms of the function itself with the function for a different term