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(A) The frequency of a spectral line is given by $v = RC \left[ \frac{1}{n_{1}^{2}} - \frac{1}{n_{2}^{2}} \right]$.For the series limit of the Lyman s

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

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(A) The frequency of a spectral line is given by $v = RC \left[ \frac{1}{n_{1}^{2}} - \frac{1}{n_{2}^{2}} \right]$.For the series limit of the Lyman series $(n_{1}=1, n_{2}=\infty)$: $v_{1} = RC \left[ 1 - \frac{1}{\infty} \right] = RC$.For the first line of the Lyman series $(n_{1}=1, n_{2}=2)$: $v_{2} = RC \left[ 1 - \frac{1}{4} \right] = \frac{3}{4} RC$.For the series limit of the Balmer series $(n_{1}=2, n_{2}=\infty)$: $v_{3} = RC \left[ \frac{1}{4} - \frac{1}{\infty} \right] = \frac{RC}{4}$.Comparing these values,we see that $v_{1} - v_{2} = RC - \frac{3}{4} RC = \frac{1}{4} RC = v_{3}$.Therefore,$v_{1} - v_{2} = v_{3}$.

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

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