Exercise 59 Page 821

We want to find the total distance the pendulum travels before it stops. Let's start by analyzing the given distances. A pendulum that is released to swing freely travels $24$ centimeters on the first swing, $18$ centimeters on the second swing and $13.5$ centimeters on the third swing. The total distance traveled by the pendulum is given by the infinite geometric series. To find the sum of this series we first need to identify its common ratio. Consider the given terms. The common ratio of our series is $0.75.$ Before we find the sum, we need to be sure that the series converges. We check this by calculating the absolute value of the common ratio. Because the absolute value of the common ratio is less than $1,$ we know that the series converges. For this series the first term is $a_1=24$ and the common ratio is $r=0.75.$ To find its sum, we will substitute these values into the formula for the sum of an infinite geometric series. The pendulum travels a total distance of $96$ centimeters.