**Solution:**

It is given that *A* and *G* are A.M. and G.M. between two positive numbers. Let these two positive numbers be *a* and *b*.

∴

From (1) and (2), we obtain

*a* + *b* = 2*A* … (3)

*ab* = *G*_{2} … (4)

Substituting the value of *a* and *b* from (3) and (4) in the identity (*a* – *b*)^{2} = (*a* + *b*)^{2} – 4*ab*, we obtain

(*a* – *b*)^{2} = 4*A*_{2} – 4*G*_{2} = 4 (*A*_{2}–*G*_{2})

(*a* – *b*)^{2} = 4 (*A* + *G*) (*A* – *G*)

From (3) and (5), we obtain

Substituting the value of *a* in (3), we obtain

Thus, the two numbers are .