Then at the first step we shade the area A0*4*(1/4)*(1/4) = A/4. And the square that remains for shading at the second step also has area A/4.

At the second step we shade 1/4 of A/4 and leave for the third step A/(4^2).

And so on, on the n-th step we shade the area A/(4^n).

It is the sum of an infinite geometric progression with the first term A/4 and the ratio 1/4. So this sum is equal to A*(1/4)/(1-1/4) = A*(1/4)/(3/4) = A*(1/3).

So the answer 1s 1/3.

4. After the first fall from the height a the ball will go up by some height and down by the same height.

As follows: a, ra (up), ra (down), r*ra (up), r*ra (down) and so on.

We ought to compute the sum of this (infinite) series. Let's group the terms: