Arithmetic series in sigma notation

what I want to do in this video is get some practice writing series in Sigma notation and I have a series in front of us right over here we have 7 plus 9 plus 11 and we keep on adding all the way up to 405 so first let's just think about what's going on here what how can we think about what happens at each successive term so we're at 7 and then we're going to 9 and then we're going to 11 it looks like we're adding 2 every time so it looks like this is an arithmetic sequence series so we add 2 and then we add 2 again and we're going to add we're going to keep adding to all the way until we get to 405 so let's think about how many times we are going to add 2 to get to 200 sorry how many times we have to add 2 to get to 405 so 405 is 7 plus 2 times what so let me write this down so if we wanted 405 is equal to 7 plus 2 times I'll just write 2 times X I just I'm just trying to figure out how many times do I have to add 2 to 7 to get to 405 and so that is going to be equal to let's see if we subtract 7 from both sides we have 398 is equal to 2x or let's see divide both sides by 2 and we get this is going to be what 199 199 is equal to X so we're essentially adding to 199 times so this is the first time we're adding 2 this is 2 times writing 2 times 1 adding 2 times 2 and here we're adding 2 times 199 to our original 7 so let's think about this a little bit so this is going to be a sum sum from so there's a couple of ways we could think about it we could think about how many times we've added 2 so we could start with us adding to 0 times the number 7 is when we haven't added 2 at all all the way to when we add to 199 times and let's think about this little bit this is going to be we could write it as 7 plus 2 times K 7 times 2 7 plus 2 times K when K equals 0 this is going to be 7 when K equals 1 it's 7 plus 2 times 1 well it's going to be 9 when K is equal to 2 it's going to be 7 plus 2 times 2 which is 11 and all the way when K is equal to 199 it's going to be 7 plus 2 times 199 which is 398 which would be 405 so that's one way that we could write it another way we could also write it as let me need to do this in a different color we could if we want to start our index at K is equal to 1 then let's see it's going to be the first term is going to be 7 plus 2 times K minus 1 times K minus 1 notice the first term works out because we're not adding 2 at all so 1 minus 1 is equal to 0 so you're just going to get 7 then when K is equal to 2 the second term we're going to add 2 one time because 2 minus 1 is 2 so that gives us that 1 and so how many total terms are we going to have here well one way to think about is I just shifted the indices up by one so we're going to go from K equals 1 to 200 and you can verify this when K is equal to 200 this is going to be 200 minus 1 which is 199 2 times 199 is 398 plus 7 is indeed 405 so when K equals 200 that is our last term here so either way these are legitimate ways of expressing this arithmetic series in using Sigma notation