 Hi, I'm Zor. Welcome to Unizor Education. We will talk about sequences. Well, let me start with a question. What's the difference between a set of all, let's say, even numbers and a sequence of all even numbers? They will be the same, basically, right? So, what's the difference and why do we use the term set in one case and the term sequence in another case? Well, the answer is very simple. Well, I asked the question, I will answer. The answer is that sequence is basically an ordered set, a set where you know what is element number one in this set, element number two, number three, etc. If you just say a set, you don't really order all the elements. So, basically, if this is a set of all even numbers, this is also a set of all even numbers. These two sets are exactly the same, because their composition is the same, their elements are exactly the same. It doesn't really matter how I write it. But if I'm talking about sequence of all even numbers, what I mean is that this is element number one, this is element number two, this is element number three, etc. If I will position it somehow differently, let's say four, two, etc., that's a completely different sequence, even if the elements are the same. So, when we talk about sequences, we talk about ordered sets. Now, what ordered means? Yeah, I just wrote number one, number two, number three. Intuitively, it's understandable. But what is it from a mathematical standpoint? It's actually a quite simple thing, because this is actually a function. This is a representation of a function. Function has domain and codomain. These are where the function arguments are taken from, and these are the values of this function. So, what is domain and codomain in this particular case? Very simple. Domain is a set of all natural numbers. And the codomain is, in this case, it's a set of all even numbers. And the function is the rule which establishes this type of correspondence. So, let's forget about the fact that sequence maybe is something different, something we don't really know anything about it. It's a known concept. It's a concept of a function with a particular fixed and pretty much defined domain where the arguments are taken from. These are natural numbers. That's what ordering actually is. Ordering means it's a rule which puts into the correspondence natural numbers and elements of our set. That's what we mean when we're saying let's order this particular set. That means let's establish a function which means the rules between these two things which establish the correspondence. So, sequence is a function with a well-defined domain which contains natural numbers. Sometimes we can consider sequences which are finite. Let's say from one defined on the numbers and the natural numbers from one to some maximum number n. This is also a sequence. This is the finite sequence. And internet sequence is when the domain is actually all the natural numbers from one to infinity. Both are quite legitimate. So, this is a sequence of three first even numbers. And their sequence numbers are one, two, and three. This is, I use, etc. here, this is a sequence of all even numbers with corresponding sequence numbers, order numbers. This is defined on a domain which contains only three first natural numbers one, two, and three. And this one contains, this domain contains all the natural numbers. Okay, fine. So, we wanted to define what sequence actually is and that's what we did. Sequence is a function with a predefined domain of either all natural numbers from one to infinity or natural numbers from one to n. So, we can consider this as a finite sequence and this is an infinite sequence. In both cases, we start from one and either go to certain maximum or we don't stop. So, we have defined sequences. Now, what's also important is to be able to express the function which sequence actually is in some kind of a convenient form like a formula, right? So, that's what we are going to do in some cases. However, first of all, I would like to warn you that it's not always possible to come up with a formula. Let's consider the following sequence. Number one is number three. Number two is 3.1. Number three is 3.14. This is number one. This is number two. This is number three. Number four is 3.141, et cetera. Now, these are expressions of number pi which is the ratio between a circumference of a circle to its diameter. Well, we used to this one, 3.14. That's kind of the most often used number for pi. But this is approximation. Pi is irrational number which means it has an infinite number of non-periodic decimal digits after the decimal point. So, basically, I can continue this sequence of these numbers indefinitely. So, I have a sequence which represents the approximation of number pi to zero, one, two, three, et cetera decimal places. There is no formula, basically, which can express this particular type of sequence. In some other cases, let's consider a simple case which I have just used, a case of even numbers. Now, this sequence, can this be expressed as a formula? Well, I can actually guess that formula looks like this. Well, indeed. Number one times two is two. Number two times two is four. Number three times two is six, et cetera. So, it looks like the value of the sequence which has a sequence number n is 2n. Now, is this a proof? No, it's not. This is just kind of verification for a few first numbers to really prove this type of thing. You really have to do it by induction or something like this. But in any case, you obviously understand that this is the formula. Another simple formula is all odd numbers. So, you have one, three, five, seven, nine, and the function would be 2n minus one. Well, indeed, if you put one as a sequence number, you would have one times two is two minus one, one. If you put two, you will have two times two is four minus one, three. Three times two, six, minus one, five. Four times two, eight, minus one, seven, et cetera. So, this is the formula for all odd numbers. So, this sequence is quite simple. As an exercise, let me put as an example, let's say we would like to find out the formula which expresses the following sequence. Sequence of all numbers which have a remainder of, let's say, five if divided by 19. So, the first number would be obviously five. Now, the second number would be 24. 24 divided by 19 would be one and five as a remainder. Next would be 43. Because it's twice 19, which is 38 plus again five. So, all these numbers have the property of if divided by 19 as an integer, they have a remainder of five. Now, can I find a formula, basically, which represents this? Well, let me just write it down. How to get five and then 24 and then 43? Obviously, the difference between them is 19, right? So, most likely, the formula contains 19 times something. Now, what? Well, if I put n for n equals to one, I will have 19 and then something. So, I will put n minus one plus five. So, for n equals to one, I have zero plus five five. For n equals to, I have two minus one is one times 19, 19 plus five, 24. For n equals three, I have three minus one is two times 19, 38 plus five, five, 43, 43. So, this is a general formula. Again, this is not a proof. To prove it, you really have to do it by induction, but it's obviously the case. Okay, now, are all the formulas are so simple? Well, no. Let me give you another example of a relatively simple sequence for which formula is really big and obvious. Okay, here it is. Let me check if this is the right thing. Yeah, it looks like it. All right. As you see, formula looks really complex. Okay? So, one plus square root of five divided by two to the nth degree minus one minus square root of five divided by two to the nth degree. And the whole thing is divided by square root of five. Complicated. Looks really scary. Well, let me basically tell you what exactly this is. There is a well-known sequence which was first suggested by Italian mathematician Fibonacci. So, it's a Fibonacci sequence. The first numbers of this sequence are one and one. After these two, each next element is formed as a sum of two previous ones. So, in this case, one plus one is two. Two previous one, one plus two is three. Two plus three is five. Three plus five is eight. Five plus eight is 13. Eight plus 13 is 21, et cetera. So, the rule is quite simple. Just take two previous numbers to get the next one. But, this is the formula. Yes, I was quite trying to surprise myself. And obviously, you can prove it by induction again. And there are some other ways to derive it, but it doesn't really matter. My point was that sometimes the formulas are really complex, which express the function connecting natural numbers, which are sequence numbers with the values, sequential values in the sequence. No matter what it is, formula might be very complex. The rule might be really very simple. But this is the sequence, legitimate sequence. And in this particular case, by the way, I do suggest you to search the web for something like Fibonacci markets and Fibonacci nature, because the Fibonacci numbers, Fibonacci sequence, is very much used in some quantitative market research. And also, it's occurring in the nature, in many, many different ways and shapes and forms. It's quite interesting actually. So, this is a few words about sequences, which I wanted to talk about. There's one more topic which remains. You know, when we're talking about functions, sometimes we are using graphs. Is the sequence as a function a function which can have a graphical representation? Well, the answer is obviously yes, because all functions which have real values as arguments and real values as elements of the color main, they can be actually represented in a graphical format when for each x you have y equals to f of x. And this point on the coordinate plane with x coordinate being an argument and y coordinate being the value of the function, this point belonging to the graph. So, graph is actually a set of points on the plane where for each point you can find out what's the argument and what's the function. Now, in our case, our argument is always a natural number, one, two, three, etc. What does it mean? It means that the graph is not some kind of a line or a curve. It's actually a composition of separate points. So, I have a point at one. I will have a graph's point at two, at three, at four, etc. So, only at these x coordinates, my graph is defined. And what's the value of my graph? Well, let's talk about, for instance, a Fibonacci sequence. So, here it is. One, one, two, three, five, eight. So, number one is one. Number two is one again. Number three is two. Number four is three. Number five is five. Let me make it longer. One, two, three, four. One, one, two, three. One, one, two, three, four. One, one, two, three, five. Five, somewhere here. And number six would be eight over there. So, this is these points, separate distinct points, and that's what's very important. These distinct points represent the graph. Graph does not exist in between these. Graph exists only as separate points, which have x coordinates, one, two, three, four, five, etc. and the corresponding values on the y-axis. So, any sequence can be expressed as a graph, but you have to always understand that this graph is not a line or a curve or anything like that. This graph contains isolated points with x coordinates, being one, two, three, four, five, etc. Okay, so basically that's it for introduction to sequences. I will spend some time during the next lectures talking about arithmetic sequences, geometric sequences, some problems, etc. This is just an introduction, so you understand from a little bit more rigorous standpoint what sequence actually is as a function, which has the domain, natural numbers. Please examine the notes for this lecture at Unisr.com. Whenever you will see some problems after some lectures are covered, try to solve them yourselves. It's very important actually if you register on the website and that allows you to take exams. To register, you actually need two people, one acting as a supervisor who enrolls you into certain courses and then you as a student will take lectures, solve problems, take exams, etc. and then supervisor again marks this particular course as completed and enrolls you in something else. That's it basically for today. Thank you very much.