 So welcome back to another screencast where we're going to look at a certain kind of recursively defined sequence that has lots of interesting properties and is really famous in mathematics called the Fibonacci sequence. Let's begin with a little concept check slash review here. So in the last video you learned about recursively defined sequences. A sequence in general is just an infinite list of numbers that's listed in order. And sometimes they have nicely defined formulas and sometimes they don't to describe the terms. Here's an example of a recursively defined sequence. It's a sequence where every term in the sequence passed a base case is determined by one or perhaps more of the terms that were already defined in the sequence like the amount of drug in the child's bloodstream we saw. This is recursively defined. We're going to give the base case here is actually consists of two terms. B zero is two, B one is five. So although we're starting the numbering at zero and one, this is still the first two terms of the sequence, two and five. Every other term in the sequence is determined by the previous two terms. Bn is defined to be bn minus one minus bn. So just for example, if I were calculating b of ten, that would be b of nine minus b of eight. So and that's for all n greater than or equal to two. So with that in mind, what's the value of b seven? See if you can compute that, pause the video and come back when you are done. So the answer here is going to be f or five. And let's quickly run through that and see what we have. We already know b zero and b one. We need to know b two and although we have to b seven. So b two would be that the subscript there on n is two. So I would be taking b one minus b zero. Those are known quantities. Those are our base case. And so I know what they are. That's five minus two, which is three. B three, just to go through this quickly, would be b two minus b one. And since I just calculated b two up here in the previous line, I can put that in three minus, and b one I'm still in the base case here. That would be three minus five, which is negative two. B four would be b three according to my rule here, minus b two. B four or b three, sorry, I just calculated up here. So that would be negative two minus whatever b two was, that would be three. So that's minus five or negative five. B five is b four minus b three. B four was just calculated, that's negative five. And b three we had already calculated up here, that's negative two. I'm subtracting it though, so this gives me negative three. B six is b five minus b four. B five I just calculated to be negative three and b four was minus five. So I'm subtracting that, that gives me a plus two. And finally, I'll do this up here in the blue part. B seven, my target here is b six minus b five. And I'd have that data now to compute it. B six was two, b five was minus three, but I'm subtracting it. So that gives me a five and that's my answer. Okay, so we can define sequences recursively with actually two terms in the base case, like so. Now we're going to introduce a new sequence now that's got a lot of interesting properties and like I said, it's really famous that is defined in such a way it's a recursively defined sequence that has two base cases. This is known as the Fibonacci sequence after the person who discovered it and actually developed this sequence of numbers you're about to see here. In an attempt to solve a problem about the breeding of rabbits, which is kind of an interesting way to have integer sequences show up. And you can read all about that in your textbook. So in the Fibonacci sequence we start with F one, the terms are called F. And F one is defined to be one. F two is also defined to be one. And F three and higher are defined to be Fn is equal to Fn minus one plus Fn minus two. Let's list out what those elements are in the Fibonacci sequences down here below. So F one is one, F two is one. F three is supposedly, according to my rule here, F two plus F one. So I'm just going to add the previous two terms, one plus one is two. F four would be F three, which is two, plus F two, which is one. And that would give me a three. Okay, so you can see the pattern now, every term in the Fibonacci sequence, except for the first two, is defined to be the sum of the previous two. So the next term, F five, is going to be three plus two. F six would be five plus three. And then there's F seven and so on and so forth. And this of course is an infinite sequence, but the pattern remains as we go through. And that is the Fibonacci sequence. The Fibonacci sequence has a lot of really cool and interesting properties that we're going to discover and explore in this section. One of my favorite ones is what happens when you start looking at ratios of Fibonacci numbers. Let's take a look at that for a second. So let's suppose I start dividing Fibonacci numbers, consecutive Fibonacci numbers into each other, like to take F two divided by F one. Well, we saw F two is one and F one is one, and that's equal to one. That's not so interesting. Let's move, let's keep going though. Let's look at F three over F two. Well, F three was equal to two, F two is equal to one, that's equal to two. Still not very interesting. F four over F three, let's see that would be F four is three and F two is, or F three is two, so that would be 1.5. Now bring this up here and let's keep working. F five divided by F four. Well, let's see, F four we know is three and F five would be two plus three. That's five. And so that gives me, well, five thirds, but I want to put this in a decimal form, just a real short decimal approximation. 1.6667 if you rounded that off. If you keep doing this, something interesting begins to happen. Okay, F six over F five would be eight over five. Eight fifths is 1.6 exactly. As you keep proceeding through this list of ratios here, the number you get begins to converge on something. This one, I'll skip the numbers here, this is 1.625. And if you keep listing out these, we're creating a new sequence here in other words. I have, here's my first term, here's second, here's the third. If I look at where these numbers are going, something interesting kind of happens. Here's the next few, 1.61538. The next one down the line is 1.61905 approximately. The next one is 1.61765 and so on. There appears to be a number being approached here that's around 1.61. If you kept doing this for a long time, you would get something about equal to 1.6180339887 and it keeps going. Okay, this number here is very important, it's called the golden ratio. This is an irrational number, it doesn't have a repeating or terminating decimal expansion, and we usually use the letter phi, the Greek letter phi, to denote it. Why is this number so interesting? Well, first of all, it exists and that's interesting enough. Second of all, if you look at certain patterns in art and architecture and nature, interesting things begin to develop. And this picture up here on the upper left, you see the Parthenon, a famous building in Athens. And the front face of the Parthenon can be inscribed in a rectangle, as you kind of see along the outline here. But that rectangle can be split up into other rectangles. And if you look at them, the ratio of the sides of these big rectangles, like that big rectangle, and the ratio of the sides to that rectangle, and this rectangle, and this rectangle, those are all the golden ratio. If you look at certain spirals in nature here, like this logarithmic spiral here, you see the rectangles up here. So the proportions of each of those sides, if you take this side length and divide it into this longer side length, and then take this rectangle and divide the long side length by the short side length, and then do the same recursively for these little rectangles that are being formed. That ratio will approach the golden ratio. If you look at the sunflower, for example, and the patterns of the seeds, pick a seed on the outside like right here and just sort of follow the spiral in and what you're seeing there is one of these spirals here. And that's got a ratio, sort of a curvature that approaches the golden ratio. So the golden ratio is a number that appears repeatedly in art and architecture and just naturally occurring phenomena. Nobody's really sure why, but some artists have actually used the golden ratio deliberately to set up their works. So it has this sort of naturally occurring symmetry and beauty to it. So I think it's pretty interesting. We're going to look in the next video in your classwork as well at some theorems about properties of the Fibonacci numbers that are also interesting. So stay tuned.