Two resistances `R_(1)andR_(2)` provides series to parallel equivalents as n/1, then the correct relationship is

Options

A. `((R_(1))/(R_(2)))^(3//2)+((R_(2))/(R_(1)))^(3//2)=n^(3//2)`
B. `((R_(1))/(R_(2)))^(3)+((R_(2))/(R_(1)))^(3)=n^(3)`
C. `((R_(1))/(R_(2)))^(2)+((R_(2))/(R_(1)))^(2)=n^(2)`
D. `((R_(1))/(R_(2)))^(1//2)+((R_(2))/(R_(1)))^(1//2)=n^(1//2)`

Explanation

To solve the problem, we need to derive the relationship between the resistances \( R_1 \) and \( R_2 \) given that the ratio of their series equivalent to their parallel equivalent is \( \frac{n}{1} \).

### Step-by-Step Solution:

1. **Understand the Definitions**:
- The series equivalent resistance \( R_s \) of two resistances \( R_1 \) and \( R_2 \) is given by:
\[ R_s = R_1 + R_2 \]
- The parallel equivalent resistance \( R_p \) is given by:
\[ \frac{1}{R_p} = \frac{1}{R_1} + \frac{1}{R_2} \implies R_p = \frac{R_1 R_2}{R_1 + R_2} \]

2. **Set Up the Ratio**:
- According to the problem, the ratio of the series equivalent to the parallel equivalent is:
\[ \frac{R_s}{R_p} = \frac{n}{1} \]
- Substituting the expressions for \( R_s \) and \( R_p \):
\[ \frac{R_1 + R_2}{\frac{R_1 R_2}{R_1 + R_2}} = n \]

3. **Simplify the Equation**:
- To simplify the left-hand side, we multiply both sides by \( R_p \):
\[ (R_1 + R_2)^2 = n \cdot R_1 R_2 \]

4. **Rearranging the Equation**:
- Rearranging gives us:
\[ (R_1 + R_2)^2 = n R_1 R_2 \]

5. **Taking the Square Root**:
- Taking the square root of both sides:
\[ R_1 + R_2 = \sqrt{n} \sqrt{R_1 R_2} \]

6. **Dividing by the Square Roots**:
- Dividing both sides by \( \sqrt{R_1 R_2} \):
\[ \frac{R_1 + R_2}{\sqrt{R_1 R_2}} = \sqrt{n} \]

7. **Expressing the Left Side**:
- The left side can be expressed as:
\[ \frac{R_1}{\sqrt{R_2}} + \frac{R_2}{\sqrt{R_1}} = \sqrt{n} \]

8. **Final Relationship**:
- Therefore, we conclude that:
\[ \left( \frac{R_1}{R_2} \right)^{1/2} + \left( \frac{R_2}{R_1} \right)^{1/2} = \sqrt{n} \]

### Conclusion:
The correct relationship derived from the given conditions is:
\[ \left( \frac{R_1}{R_2} \right)^{1/2} + \left( \frac{R_2}{R_1} \right)^{1/2} = \sqrt{n} \]