A ball is dropped from a height h on a floor. The coefficient of...

Height achieved in $n^{th}$ collision is given by

${h}_{n} = e^{2n}h$
Distance covered in first collision = $h$,

Distance covered in ${2}^{nd}$ collision = $2e^{2}h$, and so on.

Total distance covered by the ball before it comes to rest:

$s = h + 2e^{2}h + 2e^{4}h +......$

$⇒ s = h \left[1 + \left(2e^{2} + 2e^{4} + 2e^{6} +......\right)\right]$

from the sum of infinite geometric progression:

$⇒ s = h \left[1 + \dfrac{2e^{2}}{1 - e^{2}}\right]$

$⇒ s = h \left[\dfrac{1 + e^{2}}{1 - e^{2}}\right]$