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# Summation notation

## Video transcript

what I want to do in this video is introduce you to the idea of Sigma notation which will be used extensively through your mathematical career so let's just say you wanted to find a sum of some terms and these terms have a pattern so let's say you want to find the sum of the first 10 numbers so you could say 1 plus 2 Plus 3 plus and you go all the way to plus 9 plus 10 and I clearly could have even written this whole thing out but you can imagine it becomes a lot harder if you wanted to find the sum of the first hundred numbers so that would be 1 plus 2 Plus 3 plus and you would go all the way to 99 plus 100 so mathematicians said well there must let's let's find some way to sum notation instead of having to do this dot dot dot thing which you will see sometimes done so that we can more cleanly express these types of sums and that's where Sigma notation comes from so this sum up here right of right over here this first one it could be represented at AZ and Sigma this use a capital Sigma Greek letter right over here and what you do is you define an index and you could say and you can start your index at some value so let's say your index starts at one I also use I for index so let's say that I starts at one and I'm going to go to 10 so I starts at one and it goes to 10 and I'm going to sum up the eyes so how does this translate into this right over here well what you do is you start wherever the index is the index is at one set I equal to one write the 1 down and then you increment the index and so I will then be equal to 2 I is to put the 2 down and you're summing each of these terms as you go and you go all the way until I is equal to 10 all the way until I is equal to 10 so given what I just told you I encourage you to pause this video and write the Sigma notation for this sum right over here assuming you've given a go at it well this would be the sum the first term well it might be easy to just say well start at I equals 1 again I equals 1 but now we're not going to stop until I 100 and we're going to sum up we're going to sum up all of the eyes let's do another example let's imagine the sum let's imagine the sum from from we just from I equals I equals 0 to 50 of I don't know let me say PI pi I squared PI I squared what would this sum look like and once again I encourage you to pause the video and write it out I kind of expand out this sum well let's just go step by step I when I equals 0 this will be pi times 0 squared and that's clearly 0 but I'll run it out pi times 0 squared then we increase our I and we will we make sure that we haven't hit this that we are I isn't already this top boundary right over here or this top value so now we said i equals 1 pi times 1 squared so plus plus pi plus pi times 1 squared 1 times 1 squared well is one our top value right over here where we stop no so we keep going so then we go I equals 2 pi times 2 squared so plus plus pi times 2 squared pi times 2 squared I think you see the pattern here and we're just going to keep going all the way until at some point we're going to keep incrementing our eye is going to be 49 so it's going to be pi times 49 squared pi times 49 squared and then finally we increment I I equal becomes 50 and we're going to have plus pi times pi times 50 squared and then we say okay our eye is finally equal to this top boundary and now we can stop and so you can see this notation this Sigma notation for this sum was a much cleaner way of a much purer way of representing this than having to write out the entire sum but you'll see people switch back and forth between the two you