It’ll take one more move to be within the 6-inch pounce distance. Do you suppose the mouse still hasn’t caught on after 26 moves? If not, then it deserves to be pounced upon. Now, to get away from this game of cat and mouse, let me bounce to an example that confounded many readers of the first edition of this book. In that edition, I gave the example of the bouncing ball and a formula to solve it.
I only intended to illustrate the use of exponents, but the readers wanted more. They wanted to know why and how! So now you benefit from all the e-mails I received. Find the total distance that a super ball travels if it always bounces back 75 percent of the distance it fell. You dropped it from a window that’s 40 feet above a nice, smooth sidewalk.
Assume that the ball always falls straight down and returns straight up. (The theoretical is always easier than the practical.) Figure 4-2 shows you some of the first drops and bounces. If you want to find out the total distance (up and down and up and down and up . . . ) that a super ball travels in n bounces, if it always bounces back 75 percent of the distance it falls, then you want to add up all the distances — all of them! In Figure 4-2, I show you some of the first distances: the original 40 feet, then 75 percent of 40 = 30 feet, then 75 percent of 30 = 22.5 feet, and so on. The list of numbers 40, 30, 22.5, 16.875, 12.65625, and so on are part of an infinite geometric sequence.
A geometric sequence is formed when each term is found by multiplying the previous term by a particular number, called the ratio. I don’t go into all the good details, and you’ll have to trust me here, but I can tell you that the sum of all the terms (infinitely many) of this type of geometric sequence is found with a rather simple formula. The sum of the terms of an infinite geometric sequence where the ratio, r, is a number between 0 and 1, is found by dividing the first term of the sequence, a, by the difference between 1 and r: