Vertical distance of bouncing ball

It is a tricky problem from Khan Academy but seems to be worth looking at.

Since the ball travels up the same distance as it travels down (except for the first bounce), the total vertical motion is

10+12(10)+12(10)+(12)2(10)+(12)2(10)+...

=10+5+5+(12)(5)+(12)(5)+(12)2(5)+(12)2(5)+...

After the first term, the expression we have is twice the sum of an infinite geometric series with first term a=5 and common ratio r=12.

    Vertical distance of bouncing ball: