Answer
(a) $17\frac{8}{9}$ ft.
(b) $18-(\frac{1}{3})^{n-3}$
Work Step by Step
(a) The heights will form a geometric sequence as the height of a bounce will be 1/3 of the height before. We have $a_1=9, a_2=3, a_3=1, a_4=1/3, a_5=1/9, ...$ and the total distance the ball has traveled at the instant
it hits the ground the fifth time is given by (consider the bounces): $S_5=9+3\times2+1\times2+2/3+2/9=17\frac{8}{9}$ ft.
(b) The total distance the ball has traveled at the instant it hits the ground the nth time is given by
$S_n=a_1+2(a_2+a_3+a_4+ ... +a_n)=2(a_1+a_2+a_3+a_4+ ... + a_n)-a_1=2\times9\times\frac{1-(1/3)^n}{1-(1/3)}-9
=18\times\frac{3}{2}(1-(\frac{1}{3})^n)-9=18-(\frac{1}{3})^{n-3}$
