If $v_{1}$ is the frequency of the series limit of Lyman series, $v_{2}$ is the frequency of the first line of Lyman series and $v_{3}$ is the frequency of the series limit of the Balmer series, then:

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$\frac{1}{v_{1}} = \frac{1}{v_{2}} + \frac{1}{v_{3}}$

$v_{1} = v_{2} - v_{3}$

$v_{1} - v_{2} = v_{3}$

$\frac{1}{v_{2}} = \frac{1}{v_{1}} + \frac{1}{v_{3}}$

answer is A.

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Detailed Solution

For Lyman series, $v_{L} = Rc [\frac{1}{1^{2}} - \frac{1}{n^{2}}]$

where $n = 2, 3, 4, . . . . . . .$

For the series limit of Lyman series, $n = \infty$

$∴ v_{1} = Rc [\frac{1}{1^{2}} - \frac{1}{\infty^{2}}] = Rc . . . . . (i)$

For the first line of Lyman series, $n = 2$

$∴ v_{2} = Rc [\frac{1}{1^{2}} - \frac{1}{2^{2}}] = \frac{3}{4} Rc . . . . . (i i)$

For Balmer series

$v_{B} = Rc [\frac{1}{2^{2}} - \frac{1}{n^{2}}]$

where $n = 3, 4, 5, . . . .$

For the series limit of Balmer series, $n = \infty$

$∴ v_{3} = Rc [\frac{1}{2^{2}} - \frac{1}{\infty^{2}}] = \frac{Rc}{4} . . . . . (i i i)$

From equations (i), (ii) and (iii), we get

$v_{1} = v_{2} + v_{3} o r, v_{1} - v_{2} = v_{3}$

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