 Hello, and welcome to this screencast. In this screencast, we will calculate the sum of a finite geometric series. So here we see an example, and it's a series because we are adding up the terms of a sequence. And it's a geometric series because the ratio of consecutive terms in the sequence is always the same, which we'll see later on in this screencast. It is a finite geometric series because we are only adding a finite number of terms. We read this out loud as the sum from i equals 3 to 25 of 5 times 2 to the i-th power. And on the right, here we see the formula for the sum of a finite geometric series. So our job is to determine the value of the three variables in the formula so that we can apply it to our example. The variable a represents the first term of our series. Since our series starts with i equals 3, we substitute 3 into the expression for the general term of our series, 5 times 2 to the i-th power. And the result is 5 times 2 cubed, which is equal to 5 times 8. And that gives us 40. And so our result is the first term is equal to 40. And that's our value for a. Next, the variable r represents the common ratio of our series. Suppose we take one term of our series, a sub i equals 5 times 2 to the i-th power. And we divide it by the previous term, a sub i minus 1 equals 5 times 2 to the i minus 1 power. The fives are going to cancel. And then there is one more factor of 2 in the numerator than there is in the denominator. So we can cancel out all factors of 2 in that denominator and all in the numerator except just one factor left over. And the result of all of this cancellation gives us a simplified version of just 2. And since that result is always going to be the same, no matter what value we use for i. So if i is 3 or i is 10 or i is 22, we always get that ratio of 2. And so that means that our series has a common ratio of 2. And so that's our result for r. The variable n represents the number of terms that we are adding up in our series. We didn't start with i equals 1 for our first term. So we can't just use 25, the upper limit for our n value. Instead, n is equal to the upper limit, 25. And then we subtract 1 less than the lower limit. So we subtract 3 minus 1. To explain why we need to subtract 1 from that lower limit of 3, we can think of starting at i equals 3 as skipping the terms when i equals 1 and i equals 2. So we are skipping 3 minus 1 or 2 terms compared to starting at i equals 1. So our final result is 23 terms that we're adding up in our series. Now that we have the value for our three variables, we can calculate the sum of our series using the formula on the right. So the sum of our series is 40 multiplied by the quantity 1 minus 2 to the 23rd power. And then we're going to divide that by 1 minus 2. So we can simplify this now. So in the numerator, we get 40 multiplied by 1 minus this big number, 8,388,608. And we divide by negative 1. So of course, we want to simplify this still further. And so we can take that 40 and divide it by negative 1 and we get negative 40. And then we're going to multiply it by the quantity inside the parentheses. And we get negative 8,388,607. When we multiply these two negative numbers together, we get a positive result. And that positive result is the impressive 335,544,280. So this final result is the sum of our finite geometric series. The sum from i equals 3 to 25 of 5 times 2 to the i-th power. Thanks for watching.