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Worked example: arithmetic series (sum expression)

Video transcript

so we have the some negative 50 plus negative 44 plus negative 38 all the way we keep adding all the way we get to 2038 and then 2044 so see if you can pause this video and evaluate this sum so let's work through this together and let's just think about what's going on so the first term here is negative 15 then we go to negative 44 so the second term is negative 50 plus 6 and then the third term we add 6 again negative 44 plus 6 is negative 38 and we go all the way to here we keep adding 6 and to go from 2038 to 2044 to get to that last term we add 6 once again and so each successive term is just 6 more than the term before it so we are dealing right over here this is an the sum is an arithmetic series it's a sum of an arithmetic sequence each term is 6 more is a constant amount is the constant amount more than the term before that so we know how to take the sum of an arithmetic sequence we know that if we have if we're taking the sum of the Meuse in a new color just to have a little bit of variety on the screen if we're taking the sum of of the first n terms of an arithmetic sequence or if we're taking or for evaluating the first n terms of an arithmetic series I could say it's going to be the first term plus the last term divided by 2 you could view this as the average of the first and the last term's times the number of terms that we're dealing with so over here we know what our first and last terms are we know this right over here that is a 1 and this is our last term 2044 so that is our a sub n so the other question is is well what is n how many terms do we actually deal with and to think about that we just say well how many times do we have to add 6 to go from negative 50 to 2044 well 2044 minus negative 50 minus negative 50 well that's the same thing as 2044 50 or 2094 and the whole reason I calculated this is I want to figure out how far do I have to go from negative 52 2044 I have to I have to go up 50 just to get back to zero and then go up another 2044 it's have to go 50 just to get back to zero then go up another 2044 so I have to go up by 2094 so if I'm going if I'm adding six on every term how many times do I have to add six to increase by 2094 well let's just take 2094 and divide it by six to figure that out so 6 goes into 20 3 times 3 times 6 is 18 subtract 20 minus 18 is to bring down the 9 6 goes into 29 4 times 4 times 6 is 24 subtract 29 minus 24 is 5 bring down the 4 we have a 54 6 goes into 54 9 times 9 times 6 is 54 and we are done so to go from negative 52 2044 I have to add 6 to 349 times so I had it once I add it twice and then this right over here this is the 340 ninth time that I'm adding 6 so how many terms do I have now you might be tempted to say 349 terms but really have 349 plus 1 terms you have the 349 for every time you added 6 so this is the first time you added 6 second time you've added 6 all the way to the 349th time you added 6 so let me make it clear this this is actually a this is the 349th time added 6 to get to this but we haven't kind of the first term just yet so we're going to have so we have 300 then we have the first term and then we add 6 349 times so we have 350 terms in this sum so in this case n is going to be equal to 350 n is equal to 350 and so we can say the sum of the first 300 I do this enough I'll do this in green the sum of the first 350 terms is going to be equal to the average of the first and last term so negative 50 plus 2004 thirty-four over two over two times three hundred and fifty so let's see negative 50 plus two thousand forty four that's going to be what 2094 two thousand ninety four divided by two x times three fifty so let's see if I just take so this is going to be a COS right this is two hundred and ninety four times let's see 350 divided oh sorry 294 one of my my brain is not working mm this is two this is actually this is gonna be nineteen ninety-four my brain really really wasn't working a little while ago so this is going to be nineteen ninety four divided by two times three hundred and fifty and so let's see three hundred fifty divided by two is 175 so this is going to be one thousand nine hundred one thousand nine hundred ninety four ninety four times 175 which is equal to and I'll use a calculator for this one so let me get the calculator out so one thousand not nine hundred ninety four times 175 gives us three hundred forty-eight thousand nine hundred fifty three hundred forty-eight thousand nine hundred and fifty and we could express this in Sigma notation now now that we know what the N is I mean we've we found our answer this is what we were looking for but just in case you're curious we could write this as a sum from let's say K is equal to 1 to K is equal to 350 of let's see we could write this as as negative fifty negative 50 plus six times K minus one because the first term we don't want to actually add the six and then the last term we want to add the six 349 times which we saw so we're going to 49 times and there you have it that's this this arithmetic series written in as Sigma notation so hopefully hopefully you enjoy pardon my my little mental error earlier I don't know what was going on in my brain