Sequences and domain | Sequences | Algebra I | Khan Academy

The focus of this video is going to be on sequences, which you have hopefully already seen. If you don't know what a sequence is, I encourage you to review those videos on Khan Academy.

But we're going to focus on how we can generate the same sequence with different functions that have different domains. So let's just start with an example sequence. Let's say we have a sequence: it's a six. You could call that the first term; some people would call that the zeroth term. Six, and then if that's the first term, the second term is now a 12, then a 24, then a 48, and so on and so forth. As we'll see, there's multiple function definitions that could create the sequence.

One way to think about it is this: this is six times one, this is six times two, this is six times four, and this is six times eight. So it looks like each term is six times a power of two. Let me make that clear. This one right over here is six times two to the zero; that's six times one. This one over here is six times two to the first. This one over here is six times two squared, six times four. This right over here is six times two to the third.

One way to view this is if you view this as the zeroth term. We could define a function; let's call it a of n, where n is referring to our index or which term in the sequence, and it's equal to six times two, six times two to the n, where n starts at zero and then it keeps incrementing by one. So it's really all integers greater than or equal to zero. It's very important to specify that domain where n is an integer and n is greater than or equal to zero.

You could see what happens if n is not an integer. If you try to put a 1.5 here or something like that, then you're not going to get one of the terms in the sequence. If we don't start at zero, if we started at one, then this would be the first term in the sequence, which is not what we want. We want to generate the sequence that I originally wrote down.

Obviously, if you started at n equals negative one, then you're going to get a different value for your first term. So this is one way to essentially define or to create a function that generates this sequence. But as we'll see, there are other ways to do it. For example, let me do another one, and I'll do another color.

Let's say I have b of n, and let's say I want to instead of starting at n equals zero, I want to start at n equals one. So what you could do is when you input a one, this essentially becomes a zero. How do I do that? Well, I just subtract one from it. So I could say six times two to the n minus one power, where n is an integer, and n is greater than or equal to one.

Notice now when we put n equals one in here, we could maybe call this the first term we want to generate a six. So what happens? One minus one, we get that zeroth power that we want right over there, and so six times two to the zero is indeed six. Then when n is equal to two, it's six times two to the two minus one, which is just two to the first power. So it just becomes six times two, which is equal to twelve.

So notice these are different function definitions with different domains, but they're generating the exact same sequence. We could also do it recursively, and we've seen this in other videos. We can define a function recursively. We could say, "Alright, look, it looks like each of these terms in our sequence is twice the previous term."

So if we want a recursive definition for the sequence, we can define the first term, or in this case, we could say the zeroth term if we want to start at n equals zero. t of zero is equal to six, and then we could say t of n is equal to two times t of n minus one. This is going to be four, or maybe I'll write it this way, where n is an integer, and n is greater than or equal to zero.

This would also generate the sequence. When you put n equals zero here, you'll get that term. When you get n equals one, t of one is going to be two times t of one minus one. t of zero in that case would be t of—sorry, it would be two times t of zero, which is six. So two times six would get you twelve.

Now, if you wanted it so that it generates a six when n equals one, you could do it this way. Let me—you could write it actually—maybe I should have kept all of that. I'm gonna have to rewrite all of that, but you could write it this way: instead of saying t of zero is equal to six, we could write t of one is equal to six.

But now I would have to write a different domain where n still has to be an integer; n is an integer, and now instead of saying n is greater than or equal to zero, now n is greater than or equal to one. So hopefully this video hits the point home that there are multiple ways, either with a traditional, I guess you say explicit function or a recursive function like this. And even in either of those cases, you can have different domains and different function definitions that generate the same sequence, but you really have to think about the domain.