Advanced sigma notation
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Arithmetic series in sigma notation
- [Voiceover] What I want to do in this video is get some practice writing series in sigma notation and I have a series in front of us right over here. We have seven plus nine plus 11 and we keep on adding all the way up to 405. So first, let's just think about what's going on here. How can we think about what happens at each successive term? So we're at seven and then we're going to nine and then we're going to 11. It looks like we're adding two every time so it looks like this is an arithmetic series. So we add two and then we add two again and we're going to keep adding two all the way until we get to 405. So let's think about how many times we are going to add two to get to 200, sorry, how many times we have to add two to get to 405. So 405 is seven plus two times what? So let me write this down. So if we wanted 405 is equal to seven plus two times, I'll just write two times x. I'm just trying to figure out how many times do I have to add two to seven to get to 405? And so that is going to be equal to, let's see, so we subtract seven from both sides. We have 398 is equal to two x or let's see, divide both sides by two and we get this is going to be what? 199? 199 is equal to x. So we're essentially adding two 199 times. So this is the first time we're adding two. This is two times. We're adding two times one, adding two times two and here, we're adding two times 199 to our original seven. So let's think about this a little bit. So this is going be a sum, a sum from, so there's a couple of ways we could think about it. We could think about how many times we've added two so we could start with us adding two zero times, the number seven is when we haven't added two at all, all the way to when we add two 199 times. And let's think about this a little bit. This is going to be, we could write it as seven plus two times k. Seven times two, seven plus two times k. When k equals zero, this is just going to be seven. When k equals one, it's seven times two plus one. Well, it's going to be nine. When k is equal to two, it's going to be seven plus two times two which is 11. And all the way when k is equal to 199, it's going to be seven plus two times 199 which is 398 which would be 405. So that's one way that we could write it. Another way, we could also write it as, let me do this in a different color, we could, if we want to start our index at k is equal to one then let's see, it's going to be the first term is going to be seven plus two times k minus one, times k minus one. Notice, the first term works out because we're not adding two at all so one minus one is equal to zero so you're just going to get seven. Then when k is equal to two, the second term, we're going to add two one time because two minus one is two so that gives us that one. And so how many total terms are we going to have here? Well, one way to think about is I just shifted the indices up by one so we're going to go from k equals one to 200. And you can verify this. When k is equal to 200, this is going to be 200 minus one which is 199. Two times 199 is 398 plus seven is indeed 405. So when k equals 200, that is our last term here. So either way, these are legitimate ways of expressing this arithmetic series in using sigma notation.