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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 familiarize ourselves with the notion of a sequence. And all a sequence is 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 1 and I keep adding 3. So 1 plus 3 is 4. 4 plus 3 is 7. 7 plus 3 is 10. And let's say I only have these four terms right 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 3, and we keep adding 4. So we go to 3, to 7, to 11, 15. And you don't always have to add the same thing. We'll explore fancier sequences. The sequences 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 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. We could say that this right over here is the sequence 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'd 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 k's from k equals 1, 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 defining our sequence as explicitly using 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 1 to 4, 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 a plus 3. This right over here was a plus 3. This right over here is a plus 3. So whatever k is, we started at 1. And we added 3 one less than the k term times. So we could say that this is going to be equal to 1 plus k minus 1 times 3, or maybe I should write 3 times k minus 1-- same thing. And you can verify that this works. 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're going to 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 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 a 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 positive integers. Now, how would I denote this business right over here? Well, I could say that 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 1, this is a sub 2-- all the way to infinity. Or we could define it-- 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 a sub k is equaling-- so we're starting at 3. And we are adding 4 one less time. For the second term, we added 4 once. For the third term, we add 4 twice. For the fourth term, we add 4 three times. So we're adding 4 one less than the term that we're at. So it's going to be plus 4 times k minus 1. 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. That's not an attractive color. Let me write this in. 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 something 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 a sequence recursively, you want to define what your first term is, with a sub 1 equaling 1. You can define every other term in terms of the term before it. And so then we could write a sub k is equal to the previous term. So this is a sub k minus 1. So a given term is equal to the previous term. Let me make it clear-- this is the previous term, plus-- in this case, we're adding 3 every time. Now, how does this make sense? Well, we're defining what a sub 1 is. And if someone says, well, what happens when k equals 2? Well, they're saying, well, it's going to be a sub 2 minus 1. So it's going to be a sub 1 plus 3. Well, we know a sub 1 is 1. So it's going to be 1 plus 3, which is 4. Well, what about a sub 3? Well, it's going to be a sub 2 plus 3. a sub 2, we just calculated as 4. You add 3. It's 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. 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, a sub k minus 1, 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. 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, but the function for a different term.