in this lesson we're going to find the sum of the arithmetic series that we have on the screen now in order to get the answer we could simply use a calculator and add up the numbers 4 + 7 + 10 + 13 + 16 + 19 + 22 all the way to 58 but we're going to use a formula to get the answer now here is the formula we need to calculate the sum of an arithmetic series it's going to be the first term a sub 1 plus the last term a subn / 2 so it's the average of the first and last terms multiplied by the number of terms now we know the value of the first term it's a sub 1 that's four the last term a subn is 58 what we don't know in this problem is the number of terms n so that's what we got to find and we could use this formula to get the value of n a subn is equal to a sub 1 + n -1 * D now we know what a sub one is the first term is four we don't know the value of n d is the common difference to go from 4 to 7 we need to add three and to go from 7 to 10 we need to add three so the common difference is three now we already know the value of a subn a subn is 58 so we have 58 is equal to 4 plus I'm going to distribute the 3 to the N minus one so this becomes 3 nus 3 combining like terms we have 4 minus 3 which is postive 1 so 58 is equal to 3 n + 1 subtracting both sides by 1 we get 50 7 is equal to 3 n dividing both sides by 3 we get n is equal to 19 so this is the first term the last term is the 19th term so we're looking for the sum of the first 19 numbers in this series A sub 1 the first term is 4 a19 the last term is 58 / 2 * 19 terms 4 + 58 is 62 62 / 2 is 31 so the average of these two numbers is 31 so we take the average multiply by the number of terms and that's going to give us the sum of the arithmetic series and that's going to be 589 so if you were to write out all the numbers from 4 to 58 going up by three and and if you add those numbers you should get this answer 589 by the way for those of you who want access to more video related content feel free to check out the links in the description if you click on this more button you're going to see other videos relating to the video that you're currently watching and these links are separated by chapter and of course you could check out my website video- t.net where you'll get access to my video playlists final exam videos and also test prep videos so feel free to take a look at that when you get a chance now let's work on another problem so let's say we have the following arithmetic Series so this time it's going in decrease in order so what's the sum of this series from 288 to 16 with each successive term decreasing by four feel free to pause the video if you want to try this now we know the first term is 288 the last term is 16 just like before we need to calculate the value of N and so we could use this formula to do that a subn is 16 a sub 1 is 288 and D the common difference as we go from the first to the second term we need to add four I mean ne4 and going from the second to the third term we need to add ne4 so the common difference it's not positive4 but4 so what I'm going to do now is distribute the4 to n minus1 so we're going to have -4 n + 4 now 288 + 4 is 292 next I'm going to subtract both sides by 292 16 - 292 is - 276 and that's equal to -4n so divide both sides by4 will give us the value of n and n is 69 so this is the 69th term so now that we know how many many terms are in the sequence we can now calculate the sum using this formula so it's going to be the first term plus the last term divid two times the number of terms so we want to find the sum of the first 69 terms the first term is 288 the 69th term is 16 and N is 69 now 288 + 16 that's 304 304 / 2 tells us that the average of all the numbers in this sequence is going to be 152 which is the average of the first and last term so 152 * 69 is 10,480 8 and so that's how you can calculate the sum of this arithmetic series for the sake of practice I'll give you another one so let's say we have the numbers 96 and then 89 well let's put a plus plus 82 + 75 and then + 12 go ahead and calculate the sum of that arithmetic Series so this is the first term this is the last term just like before let's calculate n the number of terms so a subn is is 12 a sub 1 is 96 and the common difference so going from 96 to 89 or 89 to 82 we need to add -7 so D is7 Distributing -7 to n minus1 that's going to give us -7 n + 7 -1 * -7 is POS 7 and then we can add these two numbers 96 + 7 is 103 and now let's subtract both sides by 103 12 - 103 is 91 and then we could divide both sides by -7 and that is going to give us our n value which in this problem is 13 so 12 is the 13th term or a sub 13 so now that we have that we can calculate the sum so let's go ahead and let's use this formula s subn is a sub 1 + a subn / 2 * n so looking for the sum of the first 13 terms or the partial sum the first first term is 96 a subn or the 13th term is going to be 12 and N is 13 96 + 12 is 108 108 / 2 that gives us 54 which is the average of the first and last terms so if we take that average 54 multiply by the number of terms that will give us the sum which is 72 so that's all you need to do in order to calculate the sum of an arithmetic sequence or rather an arithmetic Series so remember the first thing you need to do is just to review use this formula to calculate n the number of terms once you find n you can plug everything into this equation to get the sum of the series