I've read that the effective nuclear charge increases down the group.
This seems incorrect. As we go down the group the number of protons increases and the shielding constant also increases. We can approximate the shielding constant with the number of electrons between the nucleus and the valence electrons. Hence it seems that the effective nuclear charge is the same within a given group.
The effective nuclear charge doesn't depend on the distance ( Shown through the formula above ). So the valence electrons are further away from the nucleus and experience the same effective nuclear charge down the group. Hence they have less attraction to the nucleus and the atomic radius increases down the group.
Is my intuition correct? Or does the effective nuclear charge increase down the group?