Geometric Series
(Northern Kentucky University)
A ball dropped from a height of 18 feet above the ground as shown rebounds to 2/3 of its previous height after each bounce so that after the first bounce, it rebounds to 12 feet, after the second bounce to 8 feet and so forth.

A. To what height does it rebound after its third bounce?
B. Find an expression for the height to which it rebounds after the nth bounce as a function of n, where n is a positive integer.
C. Find the total distance the ball has traveled at the point where it hits the ground on its nth bounce as a function of n, where n is a positive integer. For example, on the first bounce, it has traveled 18 feet; on the second bounce, 18 + 12 + 12 = 42 feet; on the third bounce, 18 + 12 + 12 + 8 + 8 = 58 feet; and so forth.

The height of the nth bounce will be and the sum of the distances traveled on the first n bounces is .
The value for the final summation is .