Bouncing Ball

A special rubber ball is dropped from the top of a wall that is sixteen feet high. Each time the ball hits the ground it bounces back only half as high as the distance it fell. The ball is caught when it bounces back to a high point of 1/32 of a foot. How many times does the ball hit the ground? Explain your reasoning.

(Source: Adapted from Mathematics Teaching in the Middle School, Nov-Dec 1994).

Extension:
A golf ball dropped onto a hard surface rebounds exactly 1/x of the height from which it falls. How far will it rebound after hitting the surface for the third time if it is initially dropped from a height of 18 feet?

(Source: Adapted from Mathematics Teaching in the Middle School, Mar-Apr 1996).