Non Collusive Cournot Duopoly model with two firms, zero costs and linear demand curve

I am reading Modern Microeconomics by Koutsoyiannis. In a Non Collusive Cournot Duopoly model with two firms, zero costs and linear demand curve.

Firm A produces half the total market demand to maximise revenue.

Further, Firm B takes A's output as given and operates on the left over demand curve eD' and produces 1/4th of output (AB).

Now Firm A in period 3 should respond by taking the leftover demand curve e'D' and produce $$\frac{1}{2}$$ of the leftover market that is $$(1 - \frac{1}{2} - \frac{1}{4})\frac{1}{2} = \frac{1}{8}th$$ of total market output.

But it is mentioned