When the ball is first dropped, it falls \(h\) feet. On the first bounce, it rebounds \(0.81h\) feet, which means on the second "drop" it must travel \(0.81h\) feet again. On the second bounce, the ball rebounds \(0.81(0.81h)=0.81^2h\) feet, and on the third drop falls the same distance. And so on.
So there are two directions to track:
\(\text{downward: }\displaystyle h+0.81h+0.81^2h+\cdots=\sum_{n=0}^\infty 0.81^nh\)
\(\text{upward: }\displaystyle 0.81h+0.81^2h+\cdots=\sum_{n=1}^\infty 0.81^nh\)
The total distance is the sum of these two:
\(\displaystyle\sum_{n=0}^\infty 0.81^nh+\sum_{n=1}^\infty 0.81^nh=h+2h\sum_{n=1}^\infty 0.81^n\)
Recall that for an infinite geometric sum, you have
\(\displaystyle\sum_{n=0}^\infty r^n=1+\sum_{n=1}^\infty r^n=\frac1{1-r}\)
provided that \(|r|<1\). So the total distance traveled by the ball is
\(h+2h\left(\dfrac1{1-0.81}-1\right)\approx9.53h\)
Starting with a height of \(h=16\) means the total distance is about 152.42 ft.
