The ball reach after bouncing four time? how many bounces does it take for the ball to reach a height of less than one inch?

After the 'n'th bounce, the ball returns to a maximum height of

               (100) times (0.75)^n .

-- After 4 bounces, the maximum height is (100) (0.75)^4 = 31.64 inches.

-- To look for the maximum height of 1 inch,     (100) (0.75)^x = 1

Let's take the log of each side:    log(100) + x log(0.75) = 0

Subtract  log(100)  from each side:      x log(0.75) = -log(100)

Divide each side by  log(0.75) :            x  =  -log(100) / log(0.75).

log(100) = 2 .  So ...    x = -2 / log(0.75)  =  -2 / -0.124939  =  16.00785    

So the ball returns to a slim hair more than 1 inch high on the 16th bounce,
and returns to definitely less than 1 inch high on the 17th bounce.