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Arithmetic Series - Example 2

In mathematics, an arithmetic series is the sum of the terms of an infinite sequence, or the sequence of sums of the terms of such a series. The term arithmetic means that the sequence of numbers is an integer sequence. The sequence can be finite or infinite.

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Okay, so this is the second example out of our arithmetic, Siri's Siri's and it says, find the some of the 1st 92 terms in the following Siri's. So we have this. The numbers negative. 18. Negative, 13 negative. Eight. Negative 32 and seven. So here's the thing. So many people think they can get away with just kind of finessing through and saying, Okay, Um okay, so the next time must be 12. Next term must be 17. Let's see how far I could get, but And in the previous example, you probably could have if it was, like 23 or it was like the seventh term, the 11th term. Not if it's the 92nd term. This is why you have to know the equations in order to be with softies. So let's find out what the equation for the some of the Siri's is. Well, first of all, we have to know what kind of a Siri's it is to know what kind of some equation we want to use because there's a separate equation for the arithmetic, some versus the geometric some. So from here 18 to negative 13, we're adding five negative 13 to negative eight. We're adding five, adding five again 95 again in 95 again So clearly the change is constant, which means it is an arithmetic Siri's Okay, so then we know that the some S f N is equal to end over two times a sub one plus ace of N. So what that means is that we're going to be adding the first term in the last term and then multiplying that by the number of terms divide by two. And that makes sense. Because, for example, if we had 2468 if we were to plug that into the equation that before, over two times the first term, which is, too, because last time, which is eight, that's 10 times to which is 20 and essentially like I described, what we're doing is taking the average of the first term plus last term. And then this because this two plus eight is 10 4 plus six, also 10, and so all of the numbers from the first and last next to first and next to last 2nd, 3rd to 1st and 3rd to the last, they should all equal the same number which is why, if you pair them up, you want to divide them. Might choose. That's kind of like the behind story of it. If you and hopefully that's gonna help you remember this better then it looks like if we were trying to fill in our information, the number of terms is 90 to 92 divided by two. The first term is negative 18, but unfortunately, we do not have our last term yet. So we have to find that by using our equation for the Siris of arithmetic or arithmetic Siri's So that's gonna be a sub an is equal to a sub one plus de times and minus one. If we're considering the explicit formula, then a sub en is gonna be a sub 92. Let's say it's gonna be the first room, which is negative 18 again, plus D, which we establish that to be five, because the Increases five plus five plus five plus five plus five, the number of terms is 90 to 90 to minus one. So then, if I do 90 to minus one, that's 91 91 times five is 4 55. I've got negative 18 plus 4 55 which is going to become 4 37. So that's gonna be my 92nd term. So I'm gonna plug that and right up here for my ace of N as 4 37. So then, if I do 4 37 plus negative 18 and then multiply that by 92 divided by two. That's gonna give me my some of our 1st 92 terms, and that's going to give me 1009 or 19,274. So this right here should be our final answer. So notice our first step was just in the kind of see what kind of a Siri's it is is an arithmetic are geometric, and we noticed that since the change was constant, it was arithmetic. From there we wrote out the equation for the arithmetic some and noticed that we don't have the last term. So we found that using our arithmetic sequence Siri's equation in our explicit form, and then plug that back in