Step-by-step Solution:

\(\mathbf{Assertion \ (A)}\): "If the A.M. and G.M. between two numbers are in the ratio \( \mathbf{m : n} \), then the numbers are in the ratio \[ \mathbf{m + \sqrt{m^2 - n^2} : m - \sqrt{m^2 - n^2}} \] Explanation:
Let the two numbers be \( a \) and \( b \). The arithmetic mean (A.M.) and geometric mean (G.M.) are given by: \[ \mathbf{A.M. = \frac{a + b}{2}, \quad G.M. = \sqrt{ab}} \] Given that the ratio of A.M. to G.M. is \( m : n \), we have: \[ \mathbf{\frac{\frac{a + b}{2}}{\sqrt{ab}} = \frac{m}{n}} \] Squaring both sides and solving, we obtain: \[ \mathbf{\frac{(a + b)^2}{4ab} = \frac{m^2}{n^2}} \] Setting \( \mathbf{\frac{a}{b} = r} \) and solving the quadratic equation, we get: \[ \mathbf{r = \frac{m + \sqrt{m^2 - n^2}}{m - \sqrt{m^2 - n^2}}} \] Thus, Assertion (A) is true.

\(\mathbf{Reason \ (R)}\): "If each term of a G.P. is raised to the same power, the resulting sequence also forms a G.P."

Explanation:
Let the original geometric progression (G.P.) be \( \mathbf{a, ar, ar^2, ar^3, \dots} \). If each term is raised to the power \( \mathbf{k} \), the new sequence is: \[ \mathbf{a^k, (ar)^k, (ar^2)^k, (ar^3)^k, \dots = a^k, a^k r^k, a^k r^{2k}, a^k r^{3k}, \dots} \] This is also a G.P. with first term \( \mathbf{a^k} \) and common ratio \( \mathbf{r^k} \). Thus, Reason (R) is true.

\(\mathbf{Relationship \ Between \ Assertion \ (A) \ and \ Reason \ (R)}\):
- Assertion (A) discusses the relationship between A.M., G.M., and two numbers.
- Reason (R) states a property of geometric progressions when each term is raised to a power.
While both statements are true, Reason (R) does not explain Assertion (A), as Assertion (A) is derived from properties of means, whereas Reason (R) is a property of geometric sequences.

Final Answer: \[ \mathbf{Both \ A \ and \ R \ are \ true, \ but \ R \ is \ not \ the \ correct \ explanation \ of \ A.} \]