Original subtitles

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In this video we're talking about how to find the sum of a series.

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When the series is given to us in summation notation.

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So this each symbol right here it's called Sigma.

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This is what indicates a mation notation and the value here to the right of it is our series.

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So this value right here to the right of this sigma notation that is the series.

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So we often call this series a suburban so we could write this as a sub N is equal to quantity and minus

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one times quantity and plus two.

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That's our series.

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And what we've been asked to do is find the sum of this series.

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When we start at any quolls once where we start with the value given to us here this is called the index.

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So when we start and end equals 1 and we count up to the value that's given to us up here which in this

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case is 8.

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Oftentimes it will be infinity and we'll talk about infinite series later but here we're going to start

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an equals one and we're going to go up to eight and equals one means the first term or when n is equal

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to 1 and equals 2 would be the second term and equals three would be the third term et cetera until

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we get up to a value of N equals 8.

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The value here at the top.

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So if we want to find the sum of that series Here's what we do.

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We start by putting an end equals 1 and then we know we're going to plug into 3 4 all the way up to

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8.

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Let's just go ahead and write those out here.

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Four five six seven and eight.

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OK so we're going to start with.

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Any equals 1.

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And we're going to plug any one into our series here a 7.

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So we're going to get is 1 minus 1 or 0.

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So we're going to get ZERO times one plus two which we know is three.

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So when we plug in N equals 1 we're going to get 0 times 3 then we're going to add to that whatever

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we get when we plug in N equals 2.

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Well here we're going to get to minus 1 or 1 and here we're going to get two plus two which is for us

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we get 1 times 4 then we're just going to keep going.

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Now we're going to plug in N equals three and we're going to do this all the way up to N equals 8.

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So plugging in three we get three minus one is two three plus two is five.

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Plugging in four we're going to get four minus one is three four plus two is six plugging in five we're

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going to get five minus one is four five plus two is seven plugging in six six minus one is five six

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plus two is eight plugging in seven.

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We're going to get seven minus one is six seven plus two is nine.

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And then plugging in eight.

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We're going to get eight minus one is seven eight plus two is ten.

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And notice here that when you expand the terms of this series from end equals one to end equals eight

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you start to see a pattern.

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So for this and minus one term you can see we have 0 1 2 3 4 5 6 and 7.

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And for this end plus 2.

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We end up with 3 4 5 6 7 8 9 10.

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So once you notice the pattern you don't have to calculate each value for each term.

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You can just follow the pattern until you get up to whatever the last term is.

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In this case and equals eight.

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So now we just want to simplify 0 times 3 is zero.

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So that's going to become 0 and go away 1 times 4 is for two times five is ten.

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Three times six is eighteen four times seven is twenty eight five times eight is forty six times nine

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is fifty four.

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And here we get 70 when we add all those together we end up with two hundred twenty four.

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So we can say that the sum of this series when the series is quantity and minus one times quantity and

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plus two the sum of the series from end equals 1 to and equals eight or the sum of the series through

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its first eight terms is 224.