The electron in a hydrogen atom makes a transition `n_(1) rarr n_(2)`, where `n_(1)` and `n_(2)` are the principle quantum numbers of the two states. Assume the Bohr model to be valid. The time period of the electron in the initial state is eight times that in the final state. the possible values of `n_(1)` and `n_(2)` are

Options

A. `n_(1)Deltan_(2)=2`
B. ` n_(1)8,n_(2)=2`
C. `n_(1)Delta,n_(2)=1`
D. `n_(1)=6, n_(2)=2`

Explanation

To solve the problem, we will use the Bohr model of the hydrogen atom and the relationship between the time period of the electron in different energy states.

### Step-by-Step Solution

1. **Understanding the Time Period in Bohr Model**:
The time period \( T \) of an electron in a hydrogen atom is given by the formula:
\[ T \propto n^3 \]
where \( n \) is the principal quantum number.

2. **Setting Up the Ratio of Time Periods**:
According to the problem, the time period of the electron in the initial state \( T_1 \) is 8 times that in the final state \( T_2 \):
\[ T_1 = 8 T_2 \]

3. **Expressing Time Periods in Terms of Quantum Numbers**:
Using the proportionality we established:
\[ T_1 \propto n_1^3 \quad \text{and} \quad T_2 \propto n_2^3 \]
Therefore, we can write:
\[ \frac{T_1}{T_2} = \frac{n_1^3}{n_2^3} \]

4. **Substituting the Known Ratio**:
Substituting \( T_1 = 8 T_2 \) into the ratio gives:
\[ 8 = \frac{n_1^3}{n_2^3} \]

5. **Rearranging the Equation**:
Rearranging the equation, we have:
\[ n_1^3 = 8 n_2^3 \]

6. **Taking Cube Roots**:
Taking the cube root of both sides results in:
\[ n_1 = 2 n_2 \]

7. **Finding Possible Values of \( n_1 \) and \( n_2 \)**:
Now we need to find pairs of \( n_1 \) and \( n_2 \) that satisfy \( n_1 = 2 n_2 \). We can check the options provided:
- **Option A**: \( n_1 = 6, n_2 = 3 \) → \( 6 = 2 \times 3 \) (Valid)
- **Option B**: \( n_1 = 8, n_2 = 2 \) → \( 8 \neq 2 \times 2 \) (Invalid)
- **Option C**: \( n_1 = 4, n_2 = 1 \) → \( 4 \neq 2 \times 1 \) (Invalid)
- **Option D**: \( n_1 = 6, n_2 = 2 \) → \( 6 \neq 2 \times 2 \) (Invalid)

8. **Conclusion**:
The only valid option is:
\[ n_1 = 6 \quad \text{and} \quad n_2 = 3 \]

### Final Answer:
The possible values of \( n_1 \) and \( n_2 \) are \( n_1 = 6 \) and \( n_2 = 3 \).