If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

Main content
Current time:0:00Total duration:7:02

Worked example: arithmetic series (sigma notation)

Video transcript

so I have a finite series here expressed in Sigma notation and I encourage you to pause the video and see if you can figure out what this evaluates to this is going to evaluate to a number so assuming you've had a go at it let's work through this together so this is the sum from k equals 1 to K equals 550 so we're going to have 550 terms here and it's the sum from k equals 1 to K equals 550 of 2k plus 50 so whenever I try to evaluate a series I like to just expand out the sum a little bit just so that I can I can get a feel for what it looks like so let's see this is going to look like when K is equal to 1 this is going to be 2 times 1 plus 50 when K is equal to 2 it's going to be 2 times 2 plus 50 when K is equal to 3 it's going to be 2 times 3 plus 50 and we're going to keep going all the way until we get to the last term when k is equal to 550 it's going to be 2 times 5 hundred and 50 plus 50 so let's see this first term is going to be it evaluates 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 11 hundred and 50 so that gives us a good feel for the sum for the series we're going to start at 52 and 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 we're going to take the sum of all of these and since each successive term we're increasing by the same amount we're increasing by 2 we're increasing by 2 we can recognize this as an arithmetic series we are increasing by the same amount each time and there is a formula for the sum of an arithmetic series and first we're just going to apply the formula but then we're going to get it a little bit of an intuitive sense for why that formula works and actually in other videos we have proved this formula but it's always good to get a sense that it doesn't you know this formula just doesn't come out of thin air so the formula for the sum of an arithmetic series so the of the first n terms is going to be the first term plus the nth term over two so it's really the it's really the arithmetic mean of the first and the last term's we could say the average in this I guess everyday language average of the first and last term's and then times the number of terms you actually have so if we were to try to apply it to this case we're trying to take the sum sum of the first 550 terms I'll do this in a new color just for kicks alright so we're going to take the sum of the first 550 terms and it's going to be equal to the first term so that's 52 plus the last term the nth term eleven hundred and fifty it's really just the average of those two the average of the first and the last term and then times the number of terms we have times 550 so what is this going to be well we could simplify this a little bit if we're going to take 550 divided by 2 this is going to be I could write this as time so actually let me just let me just simplify this in a different way so this is going to be the same thing as I could write the 52 divided well let me just add first so this is going to be 1200 and 2 over 2 all right I do that right yeah 1202 over 2 times 550 now 1200 and 2 divided by 2 is going to be 600 and 1 so this is equal to 600 and 1 times 550 and let's see I can multiply that out so let me just do 550 times 600 and 1 so 1 times 550 is 550 and then I have a 0 here but then I just have a 0 there so 0 times 550 I'm just gonna get a bunch of zeroes and then I go to the hundreds place 6 times 0 is 0 6 times 5 is 30 6 times 5 is 30 plus 3 is 33 you add it all together we get a 0 we get a 5 we get a 5 we get a zero we get a three we get a three we get three hundred thirty thousand five hundred and fifty that's what this whole thing that's what this whole thing sums up to now I I just said that will get a little bit of intuition for why we were able to 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 so let me write it I'm just going to switch colors again so we're going to have the sum of the first 550 terms which is what we just wrote over here we already said this going to be 52 plus 54 plus 56 + and we're going to keep going all the way to 1150 now I'm going to write it again the sum of the first 550 terms but I'm going to just write it in Reverse we can obviously swap the order in which we add it's going to be 1150 plus eleven fifty minus two which is eleven forty eight plus that minus two which is eleven forty six and we go all the way to the first term all the way to 52 now what I want to do is just I want to add these two sums so I'm just going to get 2 times the sum of the first 550 so if I had the two left sides I'm gonna have 2 times the sum of the first 550 terms and we do this generally when we prove this formula in previous videos but I always like to get a sense of where it comes from and so this is going to e be equal to well if I add these two terms right over here I get what I get 1200 and - that number should look familiar and then if I add these two right over here what do I get I get 1200 + 2 and then if I add I think you see where this is going and then if I add these two characters what do I get I get 1200 and 2 all the way to these last two characters you add them together when you get you get 1200 and - so how many 1200 2's do I have well I have 550 of them there are 550 of these terms so this is going to be equal to 550 times 1200 into - and so if you just wanted to solve or for this sum you just divide all the sides divided by two so you / - you / - you / - and that's exactly what we did over here 550 times 1202 divided by two so hopefully that gives you an intuition for things