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Geometric Series - Example 1

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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Okay, so this is gonna be the first example out of our geometric Siri's Siri's and says Find the 18th term of the Siri's, given the excuse, given the following information. So our terms are 0.75 3, 12 and 48. So if we're looking at the difference, um, we're going from here to here it increases by 2.25 but then it increases by nine, and then it increases by it's 2036. So clearly it's not an arithmetic sequence because the we're not adding or subtracting by the same value each time. So then we have to see if it's a geometric sequence by seeing if it multiplies by the same value each time. So in going from 0.75 to 3, we know it multiplies by four because 0.75 it's the same thing as 3/4. So if we multiply that by four, those canceled out and we're left with three going from 3 to 12 are also multiplied by 44 2012 to 48 is also multiplied by four. So we have in that indeed confirmed that it is a geometric sequence So then, for our geometric sequence, we know the equation is g sub en is geese of one times are to the end minus one power. In this situation, we know that Jesus end is a term that we're looking for, Joseph One is the very first term R is the rate. So the whatever we're multiply it by each time on the end, minus one. So n is and the end out of the n minus one is going to be, um, basically the number of terms. So they were looking for the 18th term. Let's go ahead and plug in the information that we need. So we've got the first term is 0.75 then our rate is four and then our end minus one star Ennis 18. So 18 minutes one is 17. Then we've got 0.75 times, four to the 17th power. At this point, you can just plug that into your calculator, and then I would get 1.288 times 10 to the 10th power, which makes a lot of sense because ultimately this is going to get really big because you're multiplying. So it's almost like it's increasing exponentially. It makes sense that our final number for 18 term is really, really, like, just keep in mind. Our first step was to think about whether it was an arithmetic or geometric sequence by seeing if we're multiplying by the same value each time, or if we're adding or subtracting by the same value each time, and then from there we set up our geometric equation and then filled in what we knew.