 Hi, I'm Zor. Welcome to Unizor education. I'd like to talk about another example of a sequence, relatively simple example. It's called geometric sequence or geometric progression. In a way, it's somehow analogous to arithmetic progression. It's different, but there is some analogy which you can imagine. So the previous lecture was about arithmetic progression and this one will be about geometric progression. So let me start with a definition. What is a geometric progression? First, you have to choose the first element. And you also have to choose something which is called a quotient or a factor or a multiplier. Now, all other elements of this sequence are formed sequentially, starting from one by multiplying by this quotient. So the number two would be a times q. Number two, number three would be a times q times q, which is a q squared. So you remember that arithmetic progression was adding the same number. Now, geometric progression is multiplying with the same number. Well, obviously, there are some trivial cases, like for instance, what if q is equal to zero? Well, you will get a and then we'll get zero, zero, zero, zero. Trivial fact, it's kind of geometric progression but very non-interesting. Now, another also trivial case is when the q is equal to one. So you will have a, a times one, which is a, a times one times one, which is still a. So it will be a, a, a. So the whole geometric progression would contain the same element repeated many times. So not interesting. So we will probably assume, whatever I will be talking about, I will assume that q is not equal to zero and not equal to one. And quite frankly, I don't think it's very interesting if you will put q equals to minus one, because if q is equal to minus one, you will have a and then minus a, and then a and then minus a. Again, it's not really, you know, changing. It's just two various jumping left and right, left and right, a minus a, a minus a. Trivial case. So these will probably be some kind of, you know, it's not really a restriction. It's just something which I don't want to consider quite frankly because it's very trivial. Now let's talk about the n's element of this particular progression. Well, as you have probably figured it out, to reach the n, n's element, I have to multiply a by q one, two, three, four, n minus one times. Right? To get to the number two, I have to multiply one times. Once to get to number three, I have to multiply a by q two times and to get the number n, I have to multiply a by q n minus one times. So this is supposed to be the formula. Now, again, how can we prove it? Again, by induction, obviously. Well, what are the characteristic properties of geometric regression? Element number one is a. So let's substitute n is equal to one here. You will have n which is equal to one minus one zero q to the power of zero is one. So you will have a, obviously. So that's fine. Number two is that subsequent element is equal to the previous multiplied by q. Well, indeed, if you will multiply a q n minus one by q, you will get a q to the nth degree and formula for n plus one is basically a times q. If this is n plus one, n plus one minus one, so it's n. So it's the same thing. So the property is checked which means basically we have proven that this is the formula for nth element of geometric regression. Now, I would like to analyze how values of geometric regression, how elements are changing basically as the n grows to infinity. Now, if you remember from the previous lecture about arithmetic progression, we basically proved that for difference of arithmetic progression positive, we go to the positive infinity unrestricted basically. So no matter what kind of a boundary we put ourselves, we will always overcome that boundary and we'll move forward being bigger and bigger and bigger. For negative d it's corresponding to the left, which means we are getting smaller and smaller numbers going to minus infinity. Now, what about geometric progression? Is it more or less the same? Well, yes and no. Actually, if you consider arithmetic progression, let's talk about the positive difference, d. It's always growing. No matter what's the value of d, if it's one half or if it's 25 or if it's a million, we are always moving forward sequentially step by step and eventually we will definitely overcome any boundaries. In case of geometric progression that's not exactly the case. Let's consider the positive quotient q. Is it always increasing the absolute value of initial number a? Well, if q is greater than one, then answer is yes. When do we increase? Well, again, let's just consider a is positive. If a is positive then we basically divide both sides of inequality by a and q is greater than one. If q is equal to one, then aq is greater than a, which means our elements are becoming bigger and bigger and bigger. Let's just consider for simplicity positive a and positive q, so I don't have to talk about absolute value, etc. For positive a and positive q, if q is greater than one, we are increasing values. What if q is less than one? Obviously the sign of this inequality would be different because whenever we multiply something by less than one, positive numbers, we will get smaller number. Whenever you multiply 25 by one-half, well, you will have 12 and a half, which is smaller. So it looks like for geometric progression there are two cases. In some cases the geometric progression is growing and growing indefinitely, basically. In absolute value, or if you consider only positive a and q, you can just say in values, it grows to the positive infinity. But for other q's which are less than one, still positive, the situation is different. The values are decreasing. So basically this is the property which I would like to talk about and quantify it. We would like to find out when exactly, same as we did for arithmetic progression, when exactly we will become greater than any number or smaller than any number. And again, let me just assume right now that this is the general formula for geometric progression and let's assume these three things. So the positive first member, the positive quotient, and the quotient is not equal to one. Now we have two cases from zero to one and from one and more to infinity, basically, greater than one, two cases. All right, so let's do the first one. If q is from zero to one. So I don't need to rule this. So this is my first case. Now this is when I'm saying that numbers are decreasing. They're smaller and smaller and smaller. So if you take, for instance, this is zero, this is eight, and you multiply it by q the first time, you get smaller. Let's say you multiply by one half. You multiply by one half again, you will have even smaller and smaller, et cetera, et cetera. And eventually it will become smaller than any number, whatever the number I choose, z. It will become smaller than this number, z. One millions, one quadrillions, whatever it is. If q is less than one, eventually it will be smaller. Well, let's check how it behaves and let's find out the number m in this sequence when it will become smaller than z. So we assume that z is any number again in this particular interval. It's somewhere here. And I'm looking for the case when my general member of geometric progression becomes less than z. I mean, obviously, z is greater than one. I don't really need this restriction because obviously z is greater than one, it will be the same formula. All right, so what can we do to solve this particular inequality for m? Well, obviously the first thing is to divide by positive a to get this. Secondly, we apply decimal logarithm to both sides and obviously it will be what is log a? Log q, sorry. q is from 0 to 1. And let me remind you the graph of function y equals log x. If x is from 0 to 1, logarithm is negative. If x is greater than 1, logarithm is positive. Well, if you forgot properties of logarithm and the fact that logarithm of a power is the power multiplied by the logarithm, you just have to repeat this from other lectures in algebra or internet, wherever. Okay, so basically log q is negative considering q is from 0 to 1, which means if I divide both sides of inequality by log q, my sign should be reversed. So I will get m minus 1 greater than log z over a divided by log q. And as a result, we get at 1 to both sides. So n should be greater than 1 plus log z over a divided by log q. So that's the formula. It gives me the sequence number n, after which my geometric progression is smaller than any number z, wherever the z is chosen. Okay, now let's consider a different case when q is greater than 1. So here we can say that aq n minus 1 would be smaller than z, and obviously greater than 0 because everything is positive here. Alright, fine. Now let's consider the case when q is greater than 1 and start doing exactly the same thing. We would like to find out when our element of geometric progression becomes greater than z. Okay, the first manipulation divided by a both sides is positive, so equation, inequality, sorry, retains the sign. Now we do the same thing again, logarithm. Labarithm is monotonously increasing function, which means if this is greater than this, the logarithm of this would be greater than the logarithm of this. So the logarithm of q to the n minus 1 would be n minus 1 log q greater than log z over a. Now log q is positive because again, if you remember the same graph for q greater than 1, the logarithm is positive. So you divide by positive number left and right parts of inequality retaining the inequality sign. So it remains greater. n minus 1 greater than log z over a divided by log q and as a formula, we have exactly the same thing as you see. n greater than 1 plus log z over a divided by log q. And in this case, it would be greater. So if n is greater than this number, then members of our progression would be greater than z. So this is kind of a border for a sequence number. Any number with the number greater, any number with the value greater than this would deliver the elements of our geometric sequence greater than z. And obviously, all subsequent will be greater as well. In this case, all subsequent will be smaller as well because we are multiplying by quotient q which is positive quotient, but it's less than 1. So we are decreasing the positive way. Now, I deliberately did not consider cases when a is negative or q is negative. They are basically exactly the same thing except you can always talk about instead of increasing to positive infinity, you will be decreasing to negative infinity. Instead of decreasing to zero from the positive side, you will have increasing to zero from the negative side. So from the absolute various standpoint, these two cases are exactly equivalent to each other. That's it for geometric progression. I will also talk about some of elements of the geometric progression when I will talk about series. That will be in other lecture. Meanwhile, there are a few problems on the internet on Unisor.com which are dedicated to sequences. Please try to solve them yourself and then go to lecture which is explaining basically what these solutions are. I am encouraging all the parents actually to take part in working with this particular website with their students, with their children or whoever their supervisor. It allows you to control the educational process. It allows children, students to take exams and for you, the parents, it allows you to basically make judgment about how well your student really started this particular material and considering you can repeat as many times as you want, any lecture or any exercise, any problem solving technique, whatever. So just repeat whatever you don't feel comfortable with and you'll be fine. Thanks very much. That's it for today.