For a certain atom, there are energy levels A,B,C corresponds to energy values `E_(A)ltE_(B)ltE_(C)`. Choose the correct option if `lambda_(1),lambda_(2),lambda_(3) `are the wavelength of rediations corresponding to the transition from C to B,B to A and C to A respectively.

Options

A. ` lambda_(3)=lambda_(1)+lambda_(2)`
B. `lambda_(3)=(lambda_(1)lambda_(2))/(lambda_(1)+lambda_(2))`
C. ` lambda_(1)+lambda_(2)+lambda_(3)=0`
D. ` lambda_(3)^(2)=lambda_(1)^(2)+lambda_(2)^(2)`

Explanation

To solve the problem, we need to analyze the energy transitions between the energy levels A, B, and C of a certain atom. The energy levels are given as \(E_A < E_B < E_C\). We need to find the relationship between the wavelengths of the radiation corresponding to the transitions from C to B, B to A, and C to A.

### Step-by-Step Solution:

1. **Identify the Energy Levels**:
- We have three energy levels: A, B, and C.
- The energy values are ordered as \(E_A < E_B < E_C\).

2. **Define the Wavelengths**:
- Let \(\lambda_1\) be the wavelength for the transition from C to B.
- Let \(\lambda_2\) be the wavelength for the transition from B to A.
- Let \(\lambda_3\) be the wavelength for the transition from C to A.

3. **Use the Energy-Wavelength Relationship**:
- The energy of a photon is related to its wavelength by the equation:
\[ E = \frac{hc}{\lambda} \]
- Where \(h\) is Planck's constant and \(c\) is the speed of light.

4. **Calculate the Energy Differences**:
- The energy difference for the transition from C to B is:
\[ E_{C \to B} = E_C - E_B \]
- The energy difference for the transition from B to A is:
\[ E_{B \to A} = E_B - E_A \]
- The energy difference for the transition from C to A is:
\[ E_{C \to A} = E_C - E_A \]

5. **Express Wavelengths in Terms of Energy**:
- From the energy-wavelength relationship, we can express the wavelengths as:
\[ \lambda_1 = \frac{hc}{E_C - E_B} \]
\[ \lambda_2 = \frac{hc}{E_B - E_A} \]
\[ \lambda_3 = \frac{hc}{E_C - E_A} \]

6. **Relate the Wavelengths**:
- The total energy difference from C to A can also be expressed as the sum of the energy differences from C to B and B to A:
\[ E_{C \to A} = E_{C \to B} + E_{B \to A} \]
- In terms of wavelengths, this gives us:
\[ \frac{hc}{\lambda_3} = \frac{hc}{\lambda_1} + \frac{hc}{\lambda_2} \]
- Cancelling \(hc\) from both sides, we have:
\[ \frac{1}{\lambda_3} = \frac{1}{\lambda_1} + \frac{1}{\lambda_2} \]

7. **Final Relationship**:
- Rearranging gives us the final relationship:
\[ \lambda_3 = \frac{\lambda_1 \lambda_2}{\lambda_1 + \lambda_2} \]

### Conclusion:
The correct option is that the relationship between the wavelengths is:
\[ \lambda_3 = \frac{\lambda_1 \lambda_2}{\lambda_1 + \lambda_2} \]