For the hydrogen spectrum last line of the lyman series has frequency v1 last line of lyman series of `He^(+)` ions has frequency `V_(2)` and `1^(st)` line of Lyman series of `He^(+)` ions has frequency `v_(3)` then

Options

A. `4 v_(1)=v_(2)+v_(3)`
B. `v_(1)=4v_(2)+v_(3)`
C. `v_(2)=v_(3)-v_(1)`
D. `v_(2)=v_(1)+v_(3)`

Answer

To solve the problem, we need to establish the relationship between the frequencies \( \nu_1 \), \( \nu_2 \), and \( \nu_3 \) for the hydrogen spectrum and the helium ion \( \text{He}^+ \).

### Step-by-Step Solution:

1. **Understanding the Lyman Series**:
- The Lyman series corresponds to electronic transitions where the final energy level \( n_1 = 1 \) (ground state).
- The last line of the Lyman series corresponds to transitions from \( n_2 = \infty \) to \( n_1 = 1 \).

2. **Frequency for Hydrogen (\( \nu_1 \))**:
- For hydrogen (\( Z = 1 \)):
\[ \nu_1 = R_H \cdot Z^2 \cdot c \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right) \]
\[ \nu_1 = R_H \cdot 1^2 \cdot c \left( \frac{1}{1^2} - \frac{1}{\infty^2} \right) = R_H \cdot c \]

3. **Frequency for Helium Ion (\( \nu_2 \))**:
- For the last line of the Lyman series of \( \text{He}^+ \) (\( Z = 2 \)):
\[ \nu_2 = R_H \cdot Z^2 \cdot c \left( \frac{1}{1^2} - \frac{1}{\infty^2} \right) \]
\[ \nu_2 = R_H \cdot 2^2 \cdot c = 4R_H \cdot c \]

4. **Frequency for the First Line of Helium Ion (\( \nu_3 \))**:
- For the first line of the Lyman series of \( \text{He}^+ \), the transition is from \( n_2 = 2 \) to \( n_1 = 1 \):
\[ \nu_3 = R_H \cdot Z^2 \cdot c \left( \frac{1}{1^2} - \frac{1}{2^2} \right) \]
\[ \nu_3 = R_H \cdot 2^2 \cdot c \left( 1 - \frac{1}{4} \right) = 4R_H \cdot c \cdot \frac{3}{4} = 3R_H \cdot c \]

5. **Establishing the Relationship**:
- Now we have:
\[ \nu_1 = R_H \cdot c \]
\[ \nu_2 = 4R_H \cdot c \]
\[ \nu_3 = 3R_H \cdot c \]
- We can relate these frequencies:
\[ \nu_2 = \nu_1 + \nu_3 \]
\[ 4R_H \cdot c = R_H \cdot c + 3R_H \cdot c \]
- This confirms that the relationship holds true.

### Final Answer:
The correct relationship is:
\[ \nu_2 = \nu_1 + \nu_3 \]