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Geometric Series - Example 3

A geometric series is a series of the form a + ar + ar^2 + ... + ar^(n-1). where 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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Okay, So this is the third example out of our geometric Siri's and says, given two terms in a geometric sequence find explicit formula, the recursive formula and some of the first eight terms. Mhm. So we have the terms, Ace. A four is equal to negative 12, and a sev seven is equal to negative 1 92. So with this information, the first thing I'm gonna do is plug it into our geometric Siri's formula, um, in order to find my explicit formula. So then, if we have explosive formula that says Joseph and is equal to treat someone, times are raised to the end minus one power. So then it's gonna be negative. 12, Since that's of value is Jesus one which we don't know times are which we also don't know to the fourth power. And then we can say, make another equation using the second piece of information. So negative 1 92 is equal to, um Jesus. One times are to the seven minus once power. So then I have negative. 12 is equal to trees of one times are cute and then negative. 1 92 is equal to cheese of one times are to the sixth power I'm here. I'm gonna divide both sides of the equation by our just six so that I can isolate my chews up one. And then from here, I'm going to divide both sides by are cute again. I slipped the juice of one. So now I've got that Jesus one is equal to negative 12 over our acute and Jesus one is equal to negative 1 92 divided by r 26 So then I'm going to set those equal to each other since we know that Jesus upon and trees upon are equal So then I'm gonna get negative. 12 over r Cubed is equal to negative 1 92 over our to the sixth. From here, I'm gonna multiply both sides of the equation by our to the sixth to get rid of all the denominators. So our to the six and artistic will cancel out and are cute and are to the six will cancel out with are cute. So our cubed times negative 12 is equals negative 1 92. At this point, you can divide both sides of the equation by negative 12, which is going to give me 16. So our cube is equal to 16. So then, if we, uh So you see them from here? If we consider the cube root, take the cube root of both sides, then I'm going to get, uh, 2.5, so our is equal to 2.5. So then from here, now, I know my are, So I'm gonna plug that into either one of the equations to find out what my G someone is gonna be. And then So if I do that, um, I'm going to get that negative 12 is equal to to use of one times to negative 2.5 ways to do you and minus one power. And so that's going to give me Let's see or so to the third power. Sorry, some since we're using this equation here to the third. If we raised up to the third power ups, so and then we divide both sides by negative 12. Negative 12 divided by 16 gives us that G one is equal to 3/4. That's our very first term. So then, using that first term, we could make that explicit formula which I'll put up here. So Joseph and is equal to G one, which we just found us three or four. So three or four times are, um, which we found to be negative 2.5 ways to the end, minus month power. So here's our explicit formula. So then you get rid of some of this. So then we can use that to find the recursive formula, which for the recursive that one is a little bit easier. And then it's just Joseph and is equal to the previous term. Jesup and minus one times are So then we can just say that we found the are to be negative 2.5. We can say Joseph N is Joseph n minus one times negative 2.5. So here's our recursive from them. So if you've already found your explicit your recursive, it's doesn't even require any calculations. You just plug it in and then find the formula. So then we want to know that some of the first eight terms Well, how do we do that? Well, we have to use our equation for geometric Siri's, which is going to be s a N is equal to a one times one minus are to the end of power divided by one minus are. So then, from here, we could just plug in what we know. So we know that a the first term we found that to be three or four, it was gonna be 3/4 times one minus are, which is negative. 2.5 to the end power, which is eight divided by one minus the rate, which is negative. 2.5. So then from here, if we just do that Calculation 2.75 times one minus negative. 2.5 to 8. Divided by 1 to 3.5. And that's going to give me 327.19 So this is gonna be the some off the first eight terms.