Arithmetic sum sequence formula12/28/2023 ![]() It's going to be 1150 plus 1150 minus 2, which is 1148, plus that minus 2, which is 1146, and we go all the way to the first term, all the way to 52. I'm going to write it again, the sum of the first 550 terms, but I'm going to just write it in reverse. Sum of the first 550 terms, which is what we just wrote over here, we already said this is going to be 52 plus 54 plus 56 plus, and we're going to So, let me write it, I'm just going to switch colors again. ![]() Just apply this formula, and let's just think about what the sum of the first 550 terms is, and I just wrote it down up here. Get a little bit of intuition for why we were able to ![]() That's what this whole thing, that's what this whole thing sums up to. You add it all together, we get a zero, we get a 5, we get a 5, we get a zero, we get a 3, we get a 3. So, let me just do 550 times 601, so 1 times 550 is 550, and then I have a zero here, but I just have a zero there, so zero times 550, I'm just going to get a bunch of zeros, and then I go to the hundreds place. Now, 1202 divided by 2 is going to be 601, so this is equal to 601 times 550. All right, did I do that right? Yeah, 1,202 over 2 times 550. So this is going to be the same thing as, I could write this, 52 divided, well, let me just add first. If we're going to take 550 divided by 2, this is going to be, I could write this as times, actually, let me just, let me just simply this Number of terms we have, times 550, so what is this going to be? Well, we could simplify this a little bit. All right, so we're going to take the sum of the first 550 terms, and it's going to beĮqual to the first term, so that's 52 plus the last term, the nth term, 1150, it's really just the average of those two, the average of the first and the last term and then times the Of terms you actually have, so if we were to try toĪpply it to this case, we're trying to take the sum, sum of the first 550 terms, I'll do this in a newĬolor just for kicks. Mean of the first and the last terms, you could say the average, in, I guess, everyday language, average of the first and last terms and then times the number N terms is going to be the first term plus the nth term over 2, so it's really the, it's really the arithmetic Just doesn't come out of thin air, so the formula for the sum of an arithmetic series, so the sum of the first Good to get a sense that, you know, that this formula Videos, we have proved this formula, but it's always And there is a formula for the sum of an arithmetic series, andįirst we're just going to apply the formula, but then we're going to getĪ little bit of an intuitive sense for why that formula works, and actually, in other We are increasing by the same amount each time. We can recognize this as an arithmetic series. Term, we're increasing by the same amount, we're increasing by 2, we're increasing by 2, ![]() The sum of all of these, and since each successive We're just going to keep adding 2 for each successive term, all the way until we get to 1150, and we're going to take So that gives us a good feel for this sum, for this series. This first term is going to be, it evalutes to 52 plus, this next term is 2 times 2 plus 50 is going to be 54, plus the next term, 2 times 3 is 6 plus 50 is 56, and we're going to go all the way, all the way to our last term, 2 times 550 is 1100 plus 50 is going to be 1150. Going all the way until we get to the last term, when k is equal to 550, it's going to be 2 times 550 plus 50. When k is equal to 3, it's going to be 2 times 3 plus 50. When k is equal to 2, it's going to be 2 times 2 plus 50. This is going to look like, when k is equal 1, this is going to be 2 times 1 plus 50. Out the sum a little bit just so that I can get aįeel for what it looks like, so let's see. So this is a sum from kĮquals 1 to k equals 550, so we're going to have 550 terms here and it's the sum from kĮquals 1 to k equals 550 of 2k plus 50, so whenever I try to evaluate a series, I like to just expand So assuming you've had a go at it, let's work through this together. See if you can figure out what this evaluates to. A finite series here expressed in sigma notationĪnd I encourage you to pause the video and
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