If `v_1` is the frequency of the series limit of lyman seies, `v_2` is the freqency of the first line of lyman series and `v_3` is the fequecny of the series limit of the balmer series, then

Options

A. `v_1-v_2=v_3`
B. `v_1=v_2-v_3`
C. `(1)/(v_2)=(1)/(v_1)+(1)/(v_3)`
D. `(1)/(v_1)=(1)/(v_2)+(1)/(v_3)`

Explanation

To find the relationship between the frequencies \( v_1 \), \( v_2 \), and \( v_3 \) as described in the question, we need to analyze the transitions in the hydrogen atom for the Lyman and Balmer series.

### Step-by-Step Solution:

1. **Understand the Frequencies**:
- \( v_1 \): Frequency of the series limit of the Lyman series.
- \( v_2 \): Frequency of the first line of the Lyman series.
- \( v_3 \): Frequency of the series limit of the Balmer series.

2. **Energy Transition Formula**:
The energy difference during an electron transition in a hydrogen atom can be expressed as:
\[ \Delta E = 13.6 \, Z^2 \left( \frac{1}{n_{\text{final}}^2} - \frac{1}{n_{\text{initial}}^2} \right) \]
where \( Z \) is the atomic number (for hydrogen, \( Z = 1 \)).

3. **Frequency and Energy Relation**:
According to Planck's relation, the frequency is related to the energy change by:
\[ E = h \cdot v \]
Therefore, we can express frequency in terms of energy:
\[ v = \frac{\Delta E}{h} \]

4. **Frequency of the Series Limit of Lyman Series (\( v_1 \))**:
For the series limit of the Lyman series, the transition occurs from \( n_{\text{initial}} = \infty \) to \( n_{\text{final}} = 1 \):
\[ v_1 = K \left( \frac{1}{1^2} - \frac{1}{\infty^2} \right) = K \left( 1 - 0 \right) = K \]

5. **Frequency of the First Line of Lyman Series (\( v_2 \))**:
For the first line of the Lyman series, the transition occurs from \( n_{\text{initial}} = 2 \) to \( n_{\text{final}} = 1 \):
\[ v_2 = K \left( \frac{1}{1^2} - \frac{1}{2^2} \right) = K \left( 1 - \frac{1}{4} \right) = K \left( \frac{3}{4} \right) = \frac{3K}{4} \]

6. **Frequency of the Series Limit of Balmer Series (\( v_3 \))**:
For the series limit of the Balmer series, the transition occurs from \( n_{\text{initial}} = \infty \) to \( n_{\text{final}} = 2 \):
\[ v_3 = K \left( \frac{1}{2^2} - \frac{1}{\infty^2} \right) = K \left( \frac{1}{4} - 0 \right) = \frac{K}{4} \]

7. **Establishing the Relationship**:
Now we have:
- \( v_1 = K \)
- \( v_2 = \frac{3K}{4} \)
- \( v_3 = \frac{K}{4} \)

We can relate these frequencies:
\[ v_1 - v_2 = K - \frac{3K}{4} = \frac{K}{4} = v_3 \]

8. **Final Relation**:
Thus, we conclude that:
\[ v_1 - v_2 = v_3 \]

### Conclusion:
The relationship between the frequencies is:
\[ v_1 - v_2 = v_3 \]