If you've ever had to memorize a bank account number or a phone number or even a password, you've probably had to memorize lots of numbers. But it's not enough to just know the numbers; you also have to know the order in which they come in. Well, in this video, I'm going to show you that this is exactly what a sequence is. A sequence is really just a list of numbers in a particular or specific order. So, for example, like 2, 4, 6, 8, and so on. This is a sequence. And we're going to be talking about sequences a lot in the next couple of videos. So I'm going to give you a brief introduction, and we're going to see how they're very similar to functions. We can write equations for them, and then we'll do some examples with them. So let's go ahead and get started.

The numbers in a sequence, like, for example, the 2, 4, 6, 8, are called the terms. These may also be referred to as elements or members of a sequence. And we describe them, we will label them as, you know, for example, the first term or the second term or the third term or fourth term, so on and so forth. A lot of times, what's going to happen is that your sequences will have patterns that are happening between the numbers. So, for example, 2, 4, 6, 8. It's pretty clear to see the pattern here. Each one of these numbers just increases by 2 each time. So if I wanted to figure out the 5th term in the sequence, because it's something you're very commonly going to be asked to do with these types of problems, then all we're going to have to do is just sort of add 2 to 8, and we'll get the 5th term, which is 10. Alright? Now sequences could be finite, which means that they stop after a certain value or a certain number, or they could be infinite, which means they go on forever. Most of the time, your sequences will be infinite, but sometimes you might be dealing with sequences with finite sequences. Let's just jump into our first example and see how all this works.

So here we have these 2 sequences for examples a and b, and in each one of them, we're going to find the 5th term and we're going to identify whether it's a finite or infinite sequence. Let's take a look at the first example. I've got the numbers 3, 6, 9, 12, and then some number I'm supposed to figure out, an 18. So notice how I'm supposed to figure out what the 5th term in the sequence is. Let's go ahead and look at the pattern. Just like the 2, 4, 6, 8, 10, these numbers have a pattern between them. Each one of these things increases by the same amount each time. 3, plus 3 is 6, plus 3 is 9, plus 3 is 12, so on and so forth. So, if I wanted to figure out the 5th term in the sequence, I just add 3, and I'm going to get something like 15. So that's exactly what the 5th term in the sequence is going to be. Is this finite or infinite? Well, take a look at the last term, the 18. This kind of abruptly stops, and there's nothing that goes after it, so this is a finite sequence.

Now let's take a look at the second example. We've got 1, 1 half, 1 third, 1 fourth. Many sequences will commonly include fractions, so that's perfectly fine. So what's going on here? What's the 5th number going to be? Well, if you look here, what's going on, there's a pattern between the numbers where the numerators are always 1. So no matter what happens, I'm always going to have 1 in my numerator. What's going on in the denominator? Well, if you take a look at the first term of the 1 and you rewrite it as a fraction, like 1 over 1, we can clearly see that there's a pattern with the denominators. Each one of these denominators increases by 1 each time. So the 5th term is going to be 1 over 5. You just continue on that pattern. So that's the 5th number in the sequence. Now, unlike example a, which stopped at 18, we actually have this little after the last term in the sequence, and that means that this sequence just continues going on forever. It's infinite. So whenever you see this little dot, dot, dot, that means it's an infinite sequence. Alright?

So we can see here that sequences are really just patterns of numbers. And in that regard, they're actually kind of a lot like functions. They follow specific rules, and that means that we can write equations for them. So I want to show you the difference between functions and sequences. Let's go ahead and take a look. So with a function, something like, for example, when I had, f(x)=2x, I had inputs. And when I plugged in numbers for x, which could be anything like negative numbers or decimals or even fractions and radicals, I would plug that into this formula, and I would get the outputs. I would get the f(x) or the y terms. Sequences are a little bit different because the inputs, the things that you plug into the equations, are called indexes, and we represent those letters by the letter n. So, for example, 1, 2, 3, 4, 5 are the indexes. Now these are different because these are always going to be integers. They always start at 1 and increase by 1. So I can't plug in 1.5 into this equation. I can't figure out the 1.5th term. I can only figure out the 1st, 2nd, 3rd, 4th, 5th. It's always going to be like that. We're just starting at 1 and increasing by 1. So when I plug that into an equation, like, for example, anm=2m, my inputs are going to be the ends, and the ans, the terms of the sequence, are going to be my outputs. And we represent this by a with a little subscript notation for m. So, for example, if I plug in the inputs 1, 2, 3, 4, 5 into this equation and I multiply each one of these things by 2, I'm going to get 2, 4, 6, 8, 10, which are exactly the terms that I had in the sequence at the top of the page. And these terms here can be represented by these little a's and subscripts. This the second term. This is the 3rd term, 4th term, 5th term, so on and so forth. Alright? So, really, what we can see here is that functions and sequences are very similar. You plug in numbers to calculate stuff for them. The difference really is what kinds of numbers you can plug in for functions and sequences. For functions, you can plug in anything. And so, therefore, if you were to graph this out, you would get a line. But for sequences, I can only plug in discrete values of n. 1, 2, 3, 4, 5, and that's it. Now you're never going to have to graph anything. I'm just showing these graphs here to show you the difference between functions and sequences. So now that we have a good understanding of that, let's go ahead and take a look at our second example. We're going to find the first three terms of each sequence by using the formulas that were provided here. So in example a, we have this formula, an=n2. If you were to kind of, like, sort of, sort of think about this like a function, this is actually kind of like saying f(x)=x2, and we know how to plug in numbers for this. It works the same exact way. So if I wanted to figure out the first term in sequence, all I do is I take the an=n2 and I just replace one for n anywhere everywhere I see it inside of this equation. So for the first term, this is just going to be 1 squared, and 1 squared is just 1. So what about the second term? The second term says I'm going to take the index, which is 2, and I'm going to square that. So 2 squared is 4. And for the 3rd term, a equals or or n equals 3. What I'm going to do is I'm going to take 3, and I'm going to square that. And, therefore, this 3rd term is going to be 9. So these are the first three terms of the sequence. Let's take a look at the second example. Here we have a n equals 1 over n plus 3. So let's figure out the first term. Remember, you're just going to plug in 1, 2, 3, 4, so on and so forth every time you see n in this equation. So for the first term, you're going to have 1 over this is going to be 1 +3, so this is just going to be a 1 fourth. That's going to be your term. Again, totally fine to have fractions here. So what about the second term? This is going to be 1 over, and this is going to be 2 +3. So, in other words, this is going to be 1 5th. And we're going to have 1, for sorry. For the 3rd term, you're going to do 1 over, and this is going to be 3 +3. So, in other words, this is going to be 1 over 6. And you can continue on, but these are the first three terms of the sequence. Let's take a look at our last situation or our last example. Here, what we have is the equation an=-1tothenthpower. So sometimes you can have n's and exponents, that's perfectly fine as well. Let's take a look at the first term. The first term says I'm going to take negative one, and I'm going to raise it to the positive one power. That's the n equals one power. And negative one, just by itself, is just equal to negative one. What about the second one? Well, for the second term, this is going to be negative one raised to the second power. In other words, negative one squared. So negative one squared is just going to equal positive one. Right? So the first term is negative one. The first term is positive one. You can also have even negative numbers in terms of sequences. Alright? So what about the third one? Well, a 3 is going to be negative one to the third power. This is an odd exponent, so that means that this is just going to turn into negative one again. And we're going to see that if we continue this pattern, this thing would just keep going negative one, positive one, negative one, positive one, and so on and so forth. So these are the first three terms of the sequence. Anyway, folks, so that's the basic introduction to sequences. Let's go ahead and get some practice.