Worked example: finite geometric series (sigma notation) | High School Math | Khan Academy

Let's take, let's do some examples where we're finding the sums of finite geometric series, and let's just remind ourselves in a previous video we derived the formula where the sum of the first n terms is equal to our first term times 1 minus our common ratio to the nth power all over 1 minus our common ratio.

So, let's apply that to this finite geometric series right over here. So, what is our first term and what is our common ratio? Well, and what is our n? Some of you might just be able to pick it out by inspecting this here, but for the sake of this example, let's expand this out a little bit.

This is going to be equal to 2 * 3 to the 0, which is just 2, plus 2 * 3 to the first power, plus 2 * 3 to the second power. I could write first power there, plus 2 * 3 to the 3rd power, and we're going to go all the way to 2 * 3 to the 99th power.

So, what is our first term? What is our a? Well, a is going to be 2, and we see that in all of these terms here, so a is going to be 2. What is r? Well, each successive term as k increases by one, we're multiplying by 3 again, so 3 is our common ratio.

So, that right over there, that is r. Let me make sure that we, that is a, and now what is n going to be? Well, you might be tempted to say, well, we're going up to k equal 99, maybe n is 99, but we have to realize that we're starting at k equals 0. So, there are actually 100 terms here.

Notice when k equals 0, that's our first term. When k equals 1, that's our second term. When k equals 2, that's our third term. When k equals 3, that's our fourth term. When k equals 99, this is our 100th term.

So, what we really want to find is S sub 100. So, let's write that down. S sub 100 for this geometric series is going to be equal to 2 * (1 - 3 to the 100th power) all of that over (1 - 3).

And we could simplify this. I mean at this point, it is arithmetic that you'd be dealing with, but down here you would have a negative 2, and so you'd have 2 minus (or 2 divided by -2), so that is just a negative.

And so, negative of (1 - 3 to the 100th) that's the same thing. This is equal to 3 to the 100th power minus 1, and we're done.