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Geometric Series - Overview

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

Okay, so this video is going to be about geometric Siri's. So let's first go in and talk about the explicit formula for the geometric Siri's versus the recursive, similar to how we did for the arithmetic Siri's. So the geometric Siri's has an explicit formula of Jesus. End is equal to Jesus. One times are raised to the n minus one power. So in this context, Jesup N is the value of the term. So if you're looking for, like, the fifth term, that would be the fifth term. Um, Joseph Juan is the first term because our term value is one R is the rate. So whatever you're multiplying by each time and then N is the number of terms. So, um, example of the geometric sequence would be something like 248 16, 32. Because this time, instead of adding by the same number each time, we're multiplying by the same number each time. So going from 2 to 4, we multiplied by two 48 We multiplied by 28 16. We multiplied by two and 16 to 32. We multiplied by two, so our rate of increase is constant. It's just that instead of it being a linear increase. As in as from the arithmetic Siri's, we're gonna be seeing an exponential increase even if we're multiplying by the same amount each time. So keep in mind or multiplying by the same amount each time. And that could be a ninja jury. It could be a non integer. It could be a fraction. It could be a negative value, whatever. So then it would go up like this. So then let's think about the recursive formula so recursive so similar to the arithmetic Siri's This one is just based off of the previous term. So Jesup end is gonna be the term that came before it, which is Jesus and minus one times are so essentially. What they're saying is, whatever term came before it, you multiply it by the rate to get the new term and then so on, so forth. So that is how you do the recursive Siri's. So then we also have to talk about the equation for this some of a geometric Siri's. So for the some of the geometric Siri's, we have the equation s some end is a one times one minus are to the end divided by one minus R where r is not one. This distinction right here is just made just so that we understand that they are can't be one, because otherwise, one minus one is zero and we can't have a denominator value of zero. So even here you would just plug in the values and software s a vet.