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