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Geometric Series - Example 4

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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So this is gonna be the fourth example out of our geometric, Siri's Siri's and the question says find explicit formula, the recursive formula and some of the 1st 10 terms in the following Geometric Siri's. Okay, so let's go to start out with the explicit formula. Well, the explosive formula goes in the form of Jesup n is equal to Jesup. One times are raised to the n minus one power. So currently we don't know our Jesup one. We don't know our our but we know a couple of different other things. But we have our third term and our fifth term. So let's let's think about it. So we have a third term and then we have here. Let's see, let's say you have one, which is our third term are nine, which is our fifth term. So there's only one term between which is the fourth term, which has to be, um, the same multiple. We're multiplying by the same thing to get from 3 to 4 and 45. So let's say we multiplied by X and we multiply by X and that's how we got tonight. Well, then we could just say that one Times X is gonna be equal to the fourth term and that Times X is going to be equal to the ninth term. But then we could say one times X Times X should equal the ninth term, which means X squared is equal to nine. And so if we square root both sides X is going to be plus or minus three, but sends everything is positive, you know, it has to be plus three. So from here. So that's just one of the ways that we could have gotten it. We can also plug it into the equation, create two different equations and use the system of equations to find it. But for this since the 3rd and 5th term, or relatively close enough to each other where we can use this method, this is another thing that you could do Good. Now we've just found her Are So if I called that information in now I have chiefs of end is equal to Jesus one times three race to the n minus one power. Well, at this point, if we want to find our Jesup one, we could just plug in one of these points by no one is going to be Jesup one times three to the one minus one power. Which means it's to the for certain three to the three minus one power. Since that's the third term, then that's gonna be one is equal to juice of one times three squared. And then from here, we know that one is equal to nine times cheese of one. So we're gonna divide both sides of the equation by nine in order to isolate RG someone. So then we know that Jesus one is equal to one night. From there. We can just plug in all that information into our explicit formula template to say that Joseph N is equal to 19 which is a piece of one times three ways to the n minus one power. So here is our explicit formula. Okay, so then, for our recursive formula will recursive um, just like the name sounds, it's based on the thing that comes before it. Eso it's recurring basically so n g sub n is equal to cheese up and minus one which is basically the term before times are and initially when we were looking for the are for the exclusive formula. We found that that was three. So we know are recursive formula is just Jesup End is equal to cheese of end minus one times three. So here's a recursive pharma. So now we just have to find the some of the 1st 10 terms. Let's get rid of some of this. Okay, so then if we want to look for the some of the 1st 10 terms, we have to use the equation for that. So some is gonna be the first term. Times one minus are to the ends. Power over one minus are. So let's just go ahead and plug in everything that we know the some of the first time is gonna be the first term, which we found to be one night times one minus are, which is three to the 10th power. Since that's how many terms we want over one minus 10. Okay, so then from here, if we just plug in all that information into our calculator, So three to the 10th is 59049 something. Okay, so then that gives me that the some of the 1st 10 terms is gonna become 7. 29. So again, this part is just plugging it into the calculate. So just keep in mind that with the geometric Siri's, there's a couple different ways of solving it, and we kind of use the third term to find the 4th and 5th terms, Um, and using that to find there are for this particular question.