In general, what is the effect of the number of energy levels on the radius of an atom

The more the number of energy levels in the atom, the higher the atomic radius of the atom.

It is a common observation that atomic radius increases down the group as more shells are added to atoms. This is because electrons repel each other strongly.

This strong repulsion of electrons is called screening or shielding effect and it cancels the effect of increase in nuclear charge down the group. As more shells are added, the screening of outer electrons by the inner electrons increases thereby making the outer electrons to be found at a larger distance from the nucleus. Thus the atomic radius increases accordingly.

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The number of energy levels increased atomic radius also goes to increase.

Explanation:

Atomic radius  trend along period.

As we move from left to right across the periodic table the number of valance electrons in an atom increase. The atomic size tend to decrease in same period of periodic table because the electrons are added with in the same shell. When the electron are added, at the same time protons are also added in the nucleus. The positive charge is going to increase and this charge is greater in effect than the charge of electrons. This effect lead to the greater nuclear attraction. The electrons are pull towards the nucleus and valance shell get closer to the nucleus. As a result of this greater nuclear attraction atomic radius decreases and ionization energy increases because it is very difficult to remove the electron from atom and more energy is required.

Atomic radii trend along group:

As we move down the group atomic radii increased with increase of atomic number. The addition of electron in next energy level cause the atomic radii to increased. The hold of nucleus on valance shell become weaker because of shielding of electrons thus size of atom increased.