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# 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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##### Top Educators   ##### Michael J.

Idaho State University

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Okay, so this is gonna be the first example out of our arithmetic. Siri's, uh, arithmetic. Siri's Siri's um so then the question says, find the 23rd term of the following Siris for 7 10 13. Dada, Dada, Dada. So here's the thing you should be like, always pausing the video right here trying it for yourself and then look at my explanations if you do need help. So what I noticed about this is a couple of different things. I noticed that we have from 4 to 7, it's increasing by three from 7 to 10. It's increasing right three and 10 to 13. It's increasing by three. So what that increased tells me is that first of all, the increases by positive three and second of all the increases constant were always increasing by the same number each time. So that shows that it's arithmetic arithmetic we see because the increase is the same each time we're constant. So then let's think about our explicit equation explicit to remember. The format for the explicit equation is a sub end is equal to a sub one plus n times d. So then, if we have that, let's go ahead and fill in what we know. Well, our ace of one, which just means our very first term is for so here's a sub 23 since that's what we're looking for. And then the number of terms. I'm sorry. It's actually gonna be so we're actually gonna have the end minus one times D. Okay, so then if we're plugging that information in the D is the difference, which is three and then the n The number of terms is 23 so we've got 23 minus one. So if we simplify that, we got that ace of 23 which is the 23rd term is gonna be four. Those three times 22 which three times 22 is 66 66. Post four is 70. So we know that the 23rd term in this Siri's is going to be seventies. So going back through and recapping what we did in order to get this question Well, we first saw that it was an arithmetic sequence because that's what allowed us to kind of use that equation, because otherwise, if we didn't know it was an arithmetic equation, we couldn't use that arithmetic explicit equation format. So once we saw that the increase is constant, we knew that it was arithmetic. And then from there we plugged in the information that is required out of our equation, which we need the first term. We need the difference, and we need the number of terms, and then we just solved that through. Johns Hopkins University

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##### Top Precalculus Educators   ##### Michael J.

Idaho State University