 This video will talk about arithmetic series. So as some of an arithmetic sequence, there is this great story about Goss, a famous mathematician when his very young teacher asked to sum up the terms of an arithmetic sequence and he got it immediately. And it took the other people like the rest of the hour to finally everybody get done. And he got it immediately and then there's a nice derivation of what he probably did in your book. But this is what it boils down to. We have S of n is equal to n over 2 times the first term plus the last term. So we want to find the sum here and we know that we want to sum up 29 terms. So we're really trying to find S of 29. Well I need to know A sub 1 and that's just 4 times 1 minus 1. So A sub 1 is equal to 3 and I need to know A sub n. So A sub 29 is going to be 4 times 29 minus 1 and 4 times 29 happens to be 116 and I subtract 1 that's going to be 115. And you can do it in your calculator if you wanted to, 4 times 29 minus 1. So here we go. We have 29 terms and we're going to divide that by 2 and then we're going to add up our A sub 1 which is 3 plus our A sub n which we said was 115. So we have 29 over 2 times 118 and if I simplify this a little bit so I can get rid of my fraction the 2 is going to cancel the 118 and it goes into 11 5 times with 18 left over so 59 and 29 times 59 is 1711. That's what the first sum of the first 29 terms would be 1711. Now let's try another one but I want you to notice something here. I thought I was going to highlight but it didn't. I is equal to 7. We still have A sub n is equal to n over 2 times A sub 1 plus A sub n. n is the number of terms that we're summing so that's going to be tricky. So n is going to be 20 and then if we wanted to know the difference of them that would give me 13 but if you think about that that's the difference between 20 and 7. It doesn't really include every term. We're missing one term so we have to take the difference and then add one. So n is going to be equal to 14. We have to find our A sub 7 because it's the first term that we're working with. So A sub 7 is going to be 7 minus 14 or negative 7 and then we need to know A sub 20 because that's the last term that we want to use in our sum and that's going to be 7 minus 2 times 40. I mean 20 which is 40 which is going to give me negative 33, 7 minus 40. So A sub 20 is negative 33 and we already found out that n was 14 so let's plug and check. We want to sum up 14 terms and we have 14 terms and we have to divide that by 2. The first term, it's almost like in quotes here because it's the first term I'm using from my sum so that's going to be negative 7. And then we're going to add to that our last term so that we have negative 33. So that gives us 14 divided by 2 is 7 times negative 7 plus a negative 33 would be a negative 40 and if I multiply 7 times a negative 40 I get a negative 280. So S sub 14 is equal to negative 280. What else is to verify with our calculator so we get to use our wonderful technology again. Second stat and then we do we want the sum so we do math first and then second stat over to ops with the sequence now and our sequence is 7 minus 2x and then comma the variable we use is x and then comma we're starting with the seventh term and we're ending with the twentieth term. We tell it which terms you want to use and then we press enter and we get negative 280. We just verified.