 In this video, we're gonna provide the solution for question three on the final exam for math 12-20. And in this question, we're asked to determine the convergence of the series where N ranges from one to infinity and we add together the sequence two over five to the N. If the sum turns out to be convergent, let's actually find out the number it adds up to B. And if we think it's divergent, then we can select divergent as option F. Now, in this one, the key is to recognize that we have a geometric sequence. We're adding together a geometric sequence so we get a geometric series here. And so when it comes to a geometric series, we do a very quick convergence test. The convergence, we have to look at the constant ratio. What is the number? That's gonna repeat it over and over and over again as we progress to the sequence. And we get this from the exponential expression. We get this one over five to the N. So that means our constant ratio will be one fifth. And we're looking for the base of our exponential growth in this situation. So that's a small ratio, right? The ratio is less than one. So this tells us that we are convergent. So by that information right there, we've already determined that F is not the correct answer. We have a convergent geometric series. Well, what does it add up to B? Well, what we get here for these geometric series, these things will always add up to be A over one minus R where in this situation, the R is the constant ratio, so you get one minus one fifth in the denominator. And then A is just the first term of the sequence. Take the bottom number right here and plug it in. So when you plug in one, you're gonna get two fifths. And so now we just have to simplify this fraction. We get two fifths divided by, well, one takeaway of fifth is going to be four fifths. If you're struggling with that, just multiply by the reciprocal, two fifths times five over four. The fives cancel, two goes into four, two times. And so we should be left with one half, which is option D. And so for this question, as long as we know the geometric series formula, we could apply that and be able to answer something of a similar type.