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What I want to do in this[br]video is introduce you

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to the idea of Sigma notation,[br]which will be used extensively

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through your[br]mathematical career.

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So let's just say you wanted[br]to find a sum of some terms,

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and these terms have a pattern.

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So let's say you want to[br]find the sum of the first 10

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numbers.

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So you could say 1[br]plus 2 plus 3 plus,

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and you go all the[br]way to plus 9 plus 10.

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And I clearly could have even[br]written this whole thing out,

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but you can imagine it becomes[br]a lot harder if you wanted

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to find the sum of[br]the first 100 numbers.

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So that would be 1[br]plus 2 plus 3 plus,

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and you would go all[br]the way to 99 plus 100.

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So mathematicians said, well,[br]let's find some notation,

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instead of having to do this[br]dot dot dot thing-- which

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you will see sometimes[br]done-- so that we can more

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cleanly express[br]these types of sums.

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And that's where Sigma[br]notation comes from.

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So this sum up here, right[br]over here, this first one,

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it could be[br]represented as Sigma.

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Use a capital Sigma, this[br]Greek letter right over here.

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And what you do is[br]you define an index.

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And you could start your[br]index at some value.

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So let's say your[br]index starts at 1.

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I'll just use i for index.

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So let's say that i starts at[br]1, and I'm going to go to 10.

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So i starts at 1,[br]and it goes to 10.

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And I'm going to sum up the i's.

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So how does this translate[br]into this right over here?

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Well, what you do is you[br]start wherever the index is.

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If the index is at[br]1, set i equal to 1.

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Write the 1 down, and then[br]you increment the index.

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And so i will then[br]be equal to 2.

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i is 2.

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Put the 2 down.

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And you're summing each[br]of these terms as you go.

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And you go all the way[br]until i is equal to 10.

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So given what I just[br]told you, I encourage

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you to pause this video and[br]write the Sigma notation

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for this sum right over here.

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Assuming you've[br]given a go at it,

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well, this would be the sum.

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The first term,[br]well, it might be

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easy to just say we'll[br]start at i equals 1 again.

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But now we're not going to[br]stop until i equals 100,

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and we're going to[br]sum up all of the i's.

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Let's do another example.

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Let's imagine the sum from[br]i equals 0 to 50 of-- I

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don't know, let me[br]say-- pi i squared.

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What would this sum look like?

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And once again, I encourage[br]you to pause the video

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and write it out,[br]expand out this sum.

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Well, let's just[br]go step by step.

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When i equals 0, this will[br]be pi times 0 squared.

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And that's clearly 0,[br]but I'll write it out.

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pi times 0 squared.

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Then we increase our i.

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And, well, we make sure[br]that we haven't hit this,

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that our i isn't already[br]this top boundary

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right over here[br]or this top value.

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So now we said i[br]equals 1, pi times 1

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squared-- so plus[br]pi times 1 squared.

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Well, is 1 our top value right[br]over here, where we stop?

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No.

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So we keep going.

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So then we go i[br]equals 2, pi times 2

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squared-- so plus[br]pi times 2 squared.

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I think you see[br]the pattern here.

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And we're just going to[br]keep going all the way

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until, at some point-- we're[br]going to keeping incrementing

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our i. i is going to be 49.

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So it's going to be[br]pi times 49 squared.

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And then finally we increment[br]i. i equal becomes 50,

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and so we're going to have[br]plus pi times 50 squared.

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And then we say,[br]OK, our i is finally

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equal to this top boundary,[br]and now we can stop.

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And so you can[br]see this notation,

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this Sigma notation for this[br]sum was a much cleaner way,

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a much purer way,[br]of representing this

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than having to write[br]out the entire sum.

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But you'll see people switch[br]back and forth between the two.