Energy levels A,B,C of a certain atoms corresponding to increasing values of energy level i.e., `E_(A) lt E_(B) lt E_(C)`. If `lambda_(1), lambda_(2)` and `lambda_(3)` are the wavelengths of radiations corresponding to the transitions C to B,B to A and C to A respectively which of the following statement is correct?

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)`

Answer

To solve the problem, we need to analyze the transitions between energy levels A, B, and C of an atom and relate the wavelengths of the emitted radiation during these transitions.

### Step-by-Step Solution:

1. **Identify Energy Levels**:
- The energy levels are given as \( E_A < E_B < E_C \). This means:
- \( E_A \) is the lowest energy level.
- \( E_B \) is the middle energy level.
- \( E_C \) is the highest energy level.

2. **Define Wavelengths for Transitions**:
- The transitions and their corresponding wavelengths are:
- Transition from C to B corresponds to wavelength \( \lambda_1 \).
- Transition from B to A corresponds to wavelength \( \lambda_2 \).
- Transition from C to A corresponds to wavelength \( \lambda_3 \).

3. **Energy of Transitions**:
- The energy associated with a photon can be expressed as:
\[ E = \frac{hc}{\lambda} \]
- Therefore, we can express the energies for the transitions:
- For transition C to B:
\[ E_1 = \frac{hc}{\lambda_1} \]
- For transition B to A:
\[ E_2 = \frac{hc}{\lambda_2} \]
- For transition C to A:
\[ E_3 = \frac{hc}{\lambda_3} \]

4. **Relate the Energies**:
- The total energy for the transition from C to A can be expressed as the sum of the energies from C to B and B to A:
\[ E_3 = E_1 + E_2 \]
- Substituting the expressions for \( E_1 \) and \( E_2 \):
\[ \frac{hc}{\lambda_3} = \frac{hc}{\lambda_1} + \frac{hc}{\lambda_2} \]

5. **Cancel \( hc \)**:
- Since \( hc \) is a constant, we can cancel it from both sides:
\[ \frac{1}{\lambda_3} = \frac{1}{\lambda_1} + \frac{1}{\lambda_2} \]

6. **Rearranging the Equation**:
- This can be rearranged to find a relationship between the wavelengths:
\[ \frac{1}{\lambda_3} = \frac{\lambda_1 + \lambda_2}{\lambda_1 \lambda_2} \]
- Inverting gives:
\[ \lambda_3 = \frac{\lambda_1 \lambda_2}{\lambda_1 + \lambda_2} \]

7. **Conclusion**:
- The correct statement is that the wavelength \( \lambda_3 \) is given by the formula \( \lambda_3 = \frac{\lambda_1 \lambda_2}{\lambda_1 + \lambda_2} \).

### Final Answer:
The correct statement is:
\[ \lambda_3 = \frac{\lambda_1 \lambda_2}{\lambda_1 + \lambda_2} \]