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Geometric Series - Example 2

A geometric series is a series of the form a + ar + ar^2 + ... + ar^(n-1). Here r is a non-zero constant and a is a sequence of positive numbers. It is the total of an infinite number of terms. In between successive terms, should have a constant ratio.

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Video Transcript

So this is gonna be the second example out of our geometric Siri's Siri's. And it says, given to terms, any geometric sequence find the common ratio, explicit formula and the recursive formula. Okay, so I'm gonna go in and start with explicit formula because that one if you find all that the recursive formula, the common, they should just kind of come along with it. So for the explicit formula, we know that the general formula is G s A n is G sub one times are raised to the end minus one power. So in this situation that Jesus N is the that you were looking for Jesus, one is the first term. Are is the difference. Whatever you're multiplying by each time and then N is the number of terms. Then let's use this first piece of information, um, and plug it into this formula right here. I'm going to get Jesus and is negative 1/4. Some sets like the value. We don't know that Jesus of one the first term, and we also don't know the rate, but we know that that is the fourth term. So if I simplify this, I should get negative 1/4 is equal to G sub. One times are to the third power. Well, actually, I guess in this case, we do know our first term so that we can plug that in as well. And if we do that, we should get two times are to the third power on then. So, in order, Thio, isolate are I'm going to divide both sides of the equation by two on. That's gonna give me a negative 18 is equal to our to the third power. And then from here, I'm going to go ahead and cube root both sides because that's how I'm going to get rid of that cube and isolate the arm. Once I Cuba, both sides my are is by itself, which is what I was aiming to do. And then that allows me to get negative one half. We know that because negative one half times negative one half times negative one half is equal to negative one cake. Uh huh. So then we know we have now found our common ratio, so that's gonna be negative one half. So then, if you want to write out our explicit formula, you know that's gonna be Jesup N is equal to Jesup one, which is to times negative, one half all raised to the end, or that race to the end minus one power. So here's our explicit formula. So then, for the recursive formula senior recursive formula well, that one like it says recursive means that it's basically taking something that came before it. So it's going to be in the form of G sub n is equal to G sub end minus one times the rate. So then, if we fill in what we know, it's just gonna be Jesup. End is equal to Jesup end minus one times negative one half. So here's our recursive formula. But keep in mind that with the recursive formula, in order to be able to use it, you have to know the term that came before it. So, for example, if you know like the 17th term and 18th term, you could try to find the 19th term or something like that because you're using the term that came before it. But if you have like the third term and the and you're looking for like the 20th term, you can't really use the recursive formula otherwise you'd have to be doing using the record, of course, of formula, like like 15 different times in order to get to the 20 by using the five or whatever number is that.