Arubber bal falls from a height of 10 meters on the ground and rebounces vertically to half the height that it falls from in each time it rebounces up. find the sum of the distance which the ball covered

This is a combination of two geometric sequences, one for when the ball is falling and another for when the ball is climbing.

Since 1/2 is the common ratio, the sum of the geometric sequences will be:

s=a/(1-r)  as n goes to infinity, that is why the sum is missing (1-r^n) as it becomes one as n approaches infinity.

So the sum for the falling ball is:

s=10/(1-1/2)=20

And the sum for the climbing ball is:

s=5/(1-1/2)=10

So the total distance the ball travels in total magnitude is 20+10=30 meters