After striking the floor, a certain ball rebounds (4/5)th of height  from which it has fallen. Then the total distance 

that it travels before coming to rest, If it is gently dropped from a height of 120 m is

Initially the ball falls from a height of 120m. After striking the floor, it rebounds and goes  to a height of $\frac{4}{5}\times \left(120\right)m$  Now, it falls from a height of  $\frac{4}{5}\times \left(120\right)$m and after rebounding goes to a height  of  $\frac{4}{5}\left(\frac{4}{5}\left(120\right)\right)$ m. This process is continued till the ball comes to rest 
Hence, the total distance travelled is 

$120+120\left(\frac{4}{5}\right)+120{\left(\frac{4}{5}\right)}^2+.....=\frac{120}{1-{\displaystyle \frac{4}{5}}}=600m$