A ball is dropped from a height of 10 feet and bounces. Each
bounce is [3/4] of the height of the bounce before. Thus
after the ball hits the floor for the first time, the ball rises
to a height of 10([3/4]) = 7.5 feet, and after the it
hits the floor for the second time, the ball rises to a height of
7.5([3/4]) = 10([3/4])2 = 5.625 feet.
(a)
Find an expression for the height to which the ball
rises after it hits the floor for the nth time.
(b)
Find an expression for the total vertical distance the ball
has traveled when it hits the floor for the first, second, third,
and fourth times.
(c)
Find an expression for the total vertical distance the ball
has traveled when it hits the floor for the nth
time. Express your answer in a closed form.
Hint
1 + r + r2 + …+ rn
=
1 − rn +1
1 − r
.
2.
You might think that the ball [in the previous problem]
keeps bouncing forever since it takes infinitely many bounces.