 Hello, and welcome to a screencast today about finite geometric sums. All right, geometric sum S of n is the sum of the form, let me highlight this. S of n equals a plus a times r plus a times r squared dot dot dot, so all the way up to a times r to the n minus 1. And this sum is defined by the equation a times 1 minus r to the n, all divided by 1 minus r. So a is any real number, that's usually typically called your starting value. R is a number that's typically called your common ratio. And then n is just your number of terms. Okay, so as long as you know those three pieces of information, and as long as it follows this particular pattern, then we can figure out what the sum is. All right, so part a here says find the sum of 1 plus 1 times 1 half plus 1 times 1 half squared plus 1 times 1 half cubed. Now obviously you could just throw these numbers in your calculator, it's going to give you an answer, but that's no fun, right? Why would we want to do that? All right, so does everyone agree that this follows the pattern? Okay, all right, so go ahead and pause the video for just a second, identify your a, your r, and your n. Okay, welcome back. Hopefully you identified your a to be 1, that's going to be your first term here. Your r is 1 half because that's how these terms are changing, and then your n is going to be 4 because there are four terms. All right, so I'm going to plug this into my formula then, so I'm going to end up with, so my sum, which is going to be s of 4 in this case since there are four terms, is 1 times 1 minus, my r is a half, and my n is the power of 4, and then this is all divided by 1 minus a half. Okay, so then now we just got to figure out what this number is, it's just some number. So 1, okay, that's not going to change anything. So this is going to be 1 minus 1 half to the fourth power gives us 1 sixteenth, 1 minus a half gives us a half, so this is going to end up giving us 15 sixteenths divided by a half, which is better known as 15 over 8. All right, next one we're going to look at, let's find the sum of 17 plus 17 times 2, plus 17 times 2 squared, plus 17 times 2 cubed, plus 17 times 2 to the fourth, and you can see how these are going to get more and more obnoxious. All right, so again, pause the video, figure out your a, figure out your r, figure out your n, go ahead and plug them into our formula here, and let's see what the sum is. All right, so my a is 17, my r is 2, since each time we're multiplying by 2, and my n in this case is going to be 5, okay, because there are 5 terms. So my sum, s sub 5, is going to be 17 times 1 minus 2 to the fifth, all divided by 1 minus 2. So again, this seems kind of like an obnoxious number, and it probably is, so let's see what we get. So 17, and then we're going to have 1 minus 32, all divided by negative 1, and then when I go ahead and crunch that number out, I get 527. All right, not too shabby. Okay, last example then, find the sum of 20 minus 10, plus 5 minus 2.5, plus 1.25, minus 0.625. Interesting, so you notice this one, I didn't actually give it to you in the form that we want, right? I didn't give it to you so it looks like a plus a times r plus a times r squared. This one we're actually going to have to figure out for ourselves. All right, so how do I go from 20 to negative 10? In the same way, I go from negative 10 to positive 5, hopefully in the same way I go from 5 to negative 2.5. All right, so hopefully you guys are seeing what I'm seeing. So I'm going to rewrite this so it does look more like the pattern that we saw above, 20 plus 20 times negative 1.5, right? Everyone agree? 20 times negative 1.5 is negative 10, plus 20 times negative 1.5 squared. That will give us our 5, plus 20 times negative 1.5 cubed, plus 20 times negative 1.5 to the fourth, plus 20 times negative 1.5 to the fifth. Okay, let's identify my a, so my a is 20. My r is, so how am I going from one term to the next term? I'm multiplying by negative 1.5, and then I've got six terms total. Okay, so my s sub 6 is going to look like 20 times 1 minus negative 1.5 to the sixth. And if you're doing this in your calculator, it's very important that you put that negative 1.5 in parentheses, just like I did up here above. Okay, and then minus, or over, 1 minus a negative 1.5. Okay, so I'm going to have 20, and then 1 minus, so a negative 1.5 to the sixth is going to give me a positive, and it's going to be 1 over 64. And then that's going to be divided by 3.5, right? Because 1 minus a negative 1.5. So then when I go to figure out this fraction here, that's going to end up giving me 105 over 8. Okay, so with geometric sums, you just never know if they're going to give it to you in the form that actually looks like a geometric sum, or if you're just going to have to be on your toes and be able to figure that out. All right, thank you for watching.