The energies of energy levels A, B and C for a given atom are in the sequence `E_(A)ltE_(B)ltE_(C)`. If the radiations of wavelength `lambda_(1), lambda_(2)` and `lambda_(3)` are emitted due to the atomic transitions C to B, B to A and C to A respectively then which of the following relations is correct :-

Options

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

Explanation

To solve the problem, we need to analyze the energy transitions between the energy levels A, B, and C, and how they relate to the emitted wavelengths of radiation.

### Step-by-Step Solution:

1. **Identify Energy Levels**:
- We have three energy levels: A, B, and C.
- The relationship between the energies is given as \( E_A < E_B < E_C \).

2. **Identify Transitions**:
- The transitions and their corresponding wavelengths are:
- Transition from C to B emits radiation of wavelength \( \lambda_1 \).
- Transition from B to A emits radiation of wavelength \( \lambda_2 \).
- Transition from C to A emits radiation of wavelength \( \lambda_3 \).

3. **Energy Change for Each Transition**:
- The energy change for each transition can be expressed using the formula:
\[ E = \frac{hc}{\lambda} \]
- Therefore, for each transition, we can write:
- For C to B:
\[ E_C - E_B = \frac{hc}{\lambda_1} \]
- For B to A:
\[ E_B - E_A = \frac{hc}{\lambda_2} \]
- For C to A:
\[ E_C - E_A = \frac{hc}{\lambda_3} \]

4. **Combine Energy Changes**:
- The total energy change from C to A can be expressed as:
\[ E_C - E_A = (E_C - E_B) + (E_B - E_A) \]
- Substituting the expressions for energy changes:
\[ E_C - E_A = \left( \frac{hc}{\lambda_1} + \frac{hc}{\lambda_2} \right) \]

5. **Equate the Two Expressions**:
- Now we can equate the two expressions for \( E_C - E_A \):
\[ \frac{hc}{\lambda_3} = \frac{hc}{\lambda_1} + \frac{hc}{\lambda_2} \]
- Cancel \( hc \) from both sides:
\[ \frac{1}{\lambda_3} = \frac{1}{\lambda_1} + \frac{1}{\lambda_2} \]

6. **Final Relation**:
- Rearranging gives us the required relationship:
\[ \frac{1}{\lambda_3} = \frac{1}{\lambda_1} + \frac{1}{\lambda_2} \]
- This can also be expressed as:
\[ \lambda_3 = \frac{\lambda_1 \lambda_2}{\lambda_1 + \lambda_2} \]

### Conclusion:
The correct relation is:
\[ \frac{1}{\lambda_3} = \frac{1}{\lambda_1} + \frac{1}{\lambda_2} \]