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Sal explains sequences and gives examples of different sequences, defined either explicitly or recursively. Created by Sal Khan.
Video transcript
What I want to do in this video is to 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. Where, let's say I start at one and I keep adding three, so one plus three is four four plus three is seven, seven plus three is ten. And let's say I only have these four terms over here so this one we would call a finite sequence. I could also have an infinite sequence so, an example of an infinite sequence let's say, we start at three and we keep adding four so, go to three, to seven, to eleven, fifteen. And you don't always have to add the same thing we'll explore fancier sequences one, the sequences where you keep adding the same amount we call these arithmetic sequences which we will also, explore in more of a 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 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, that this right over here is the sequence, a sub k, for k is going from one to four, is equal to this right over here. So when we look at it this way, we can look at each of these as a terms in the sequence and this right over here would be the first term we would call that a sub one, this right over here would be the second term, we call it a sub two, I think you get the picture. a sub three, this right over here is a sub four. So this just says all of the a sub k's from k equals one from our first term all the way to the fourth term. Now I could also define it by not, 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 sequence, I could define it as a sub k from k equals one to four 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 one we get one when k is two, we get four, when k is three, we get seven so see when k is three we added three twice let me make it clear. So this was plus three, this right over here was a plus three this right over here, is a plus three. So whatever k is, we started at one and we added three one less than the k term times. So we could say that this is going to be equal to one, plus k minus one times three or maybe I should write three time k minus one, same thing. Three times k, minus one. And you can verify that this works so you can verify that this works. If k is equal to one, you're gonna get one minus one is zero and so, a sub one is going to be one, if k is equal to two you're going to have one plus three which is four. If k is equal to three, you get three times two plus one is seven so it works out. So this is one way to explicitly define our sequence with kind of this function notation. I want to make it clear, I have essentially defined a function here. If I wanted more traditional function notation I could've written a of k, or k is the term that I care about a of k is equal to one plus three times k minus one. This is essentially a function where an allowable input the domain is restricted to positive integers. Now, how would I denote this business right over here. Well, I could say this is equal to, and people tend to use a but 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 our first term so this is a sub one, this is a sub two, all the way to infinity. Or we could define if we wanted to define an explicitly as a function we could write the sequence as a sub k, where k starts at the first term and goes to infinity, with a sub k is equaling, so we're starting at three and we are 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 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. This is an explicit function. And so you might say well what's another way of defining these functions. Well we can also define it, especially with something like this 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 one, and going to four 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 one, equaling one. And then you can define every other term in the terms of the term before it and so we could write a of k plus, or let me write it this way a sub k is equal to the previous term so this is a sub k minus one, so the given term is equal to the previous term let me make it clear, this is the previous term, plus the previous term plus in this case 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 one is one, so it's going to be one plus three, which is four. Well what about a sub three, well it's going to be a sub two plus three a sub two we just calculate it as four, you add three it's going to be seven. 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 one, and I'm just going to add three 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 at the first term going to infinity with our first term, a sub one is going to be three now and every successive term a sub k, is going to be the previous term, a sub k minus one plus four. And once again you start at three, and then if you want the second term it's going to be the first term plus four, it's going to be three plus four you get to seven and you keep adding four, so both of these, this right over here is a recursive definition. We started with kind of a base case, and then every term is defined in terms of the terms before it or in terms of the function itself, but the function for a different term.