Advanced sigma notation
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Partial sums: term value from partial sum
- [Instructor] We're told that the nth partial sum of the series from n equals one to infinity of a sub n is given by. And so the sum of the first n terms is n squared plus one over n plus one. And they want us to figure out, what is the actual seventh term? And, like always, pause this video and see if you can figure it out on your own before we work through it together. All right, so, one way to think about it is: a sub seven, let's think about how that relates to different sums. So, if we have a sub one, plus a sub two, I'll just go all the way, a sub three, plus a sub four, plus a sub five, plus a sub six, plus a sub seven. So, if I were to sum all of these things together, that--this entire sum--that would be s sub seven. And if I wanted to figure out a sub seven, well I could subtract from that. I could subtract out the sum of the first six terms. I could subtract out s sub six. So, once again, what am I doing here, what is my strategy? I know the formula for the sum of the first n terms. I can use that to say, okay, I can figure out the sum of the first seven terms. That's gonna be the sum of all of these. And then I can use that same formula to figure out the sum of the first six terms, and the difference between the two, that's going to be our a sub seven. So, another way of saying what I just said is that a sub seven is going to be the sum of the first seven terms, minus the sum of the first six terms. And if you were doing this problem on your own you wouldn't have to write it out this way, I just wrote it out this way, hopefully making this statement a little bit more intuitive. Well, what is this going to be? Well, s sub seven, the sum of the first seven terms, we just, wherever we see an n, we replace it with a seven. So it's going to be seven squared plus one, over seven plus one and from that we are going to subtract s sub six, the sum of the first six terms. Well, that's going to be six squared plus one, or six plus one. And from here we just have to do a little bit of arithmetic. So, this is going to be seven squared plus one, this is 49 plus one so that is 50 over eight. And this is six squared plus one, that's 37, over seven. So, we want to find a common denominator between eight and seven. That would be 56. So this is going to be something over 56, minus something else over 56. Now, to go from eight to 56 I multiply by seven, so I need to multiple the numerator by seven as well. 50 times seven is 350. And then, this second fraction, I multiple the denominator by eight to get to 56, so I have to multiply 37 times eight. Let's see 37 times eight is going to be 240 plus 56, so that is 296. And so this is going to be equal to-- so I have a denominator of 56, 350 minus 296 is 54. So, it's 54/56. And if we wanted to reduce this a little bit, before we rewrite it maybe in a simpler form, we're rewriting the same value. This would be, let's see, could we write it as 27 over 28? Let's see, is that about? Yep, that's about as simplified as we can get. There you go, that's what a sub seven is, it's 27/28. The difference between the sum of the first seven terms, and the sum of the first six terms.