Let a_n be the total vertical distance that the rubber ball travels after the nth bounce, but only going down. Since the ball is dropped from 60 feet, a_1= 60. After each bounce the ball can bounce back to 23 of the previous height, so a_n is a geometric sequence and its common ratio is r= 23.
a_n - geometric sequence
a_1= 60, r= 2/3
The total vertical distance that ball travels going down is represented by the following sum of an infinite geometric series.
60+ 60( 2/3)+ 60( 2/3)^2+ 60( 2/3)^3+...
Since the common ratio r= 23 satisfies the inequalities - 1< r<1, we can use the formula for the sum of an infinite geometric series.
S=a_1/1- r
Let's substitute values for a_1 and r. Then, we will find S.
The total vertical distance that the ball travels only going down is S_(↓)= 180 ft. Next, we will calculate the total vertical distance that the ball travels only going up, S_(↑). Notice that S_(↓) and S_(↑) differ only in the first fall of the ball, a_1= 60.
