Explanation

(a) Particle falls from height $h$ then formula for height covered by it in $nth$ rebound is given by
$h_{n} = he^{2n}$
Where $e =$ coefficient of restitution, $n =$ No. of rebound
Total distance travelled by particle before rebounding has stopped
$H = h + 2h_{1} + 2h_{2} + 2h_{3} + 2h_{4} + \ldots$
$= h + 2he^{2} + 2he^{4} + 2he^{6} + 2he^{8} + \ldots$
$= h + 2h(e^{2} + e^{4} + e^{6} + e^{8} + \ldots)$
$= h + 2h\left\lbrack \frac{E^{2}}{1 - e^{2}} \right\rbrack = h\left\lbrack 1 + \frac{2e^{2}}{1 - e^{2}} \right\rbrack = h\left( \frac{1 + e^{2}}{1 - e^{2}} \right)$