Let $$x, y, z$$  be three positive prime numbers. The progression in which $$\sqrt x ,\sqrt y ,\sqrt z $$   can be three terms (not necessarily consecutive) is

Solution :
If in A.P., $$\sqrt y = \sqrt x + \left( {n - 1} \right)d$$     and $$\sqrt z = \sqrt x + \left( {m - 1} \right)d$$
$$\therefore \,\,\frac{{\sqrt y - \sqrt x }}{{\sqrt z - \sqrt x }} = \frac{{n - 1}}{{m - 1}},$$     a rational number. As $$x, y, z$$  are prime, $$\frac{{\sqrt y - \sqrt x }}{{\sqrt z - \sqrt x }}$$   is irrational.
∴ irrational = rational (absurd). So, $$\sqrt x ,\sqrt y ,\sqrt z $$   are not in A.P.
Similarly, they are not in G.P. or H.P.