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(A) The series limit of the Lyman series corresponds to the transition from $n = \infty$ to $n = 1$. Using the Rydberg formula $\frac{1}{\lambda} = R(

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$\frac{1}{\lambda_1} - \frac{1}{\lambda_2} = \frac{1}{\lambda_3}$

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(A) The series limit of the Lyman series corresponds to the transition from $n = \infty$ to $n = 1$. Using the Rydberg formula $\frac{1}{\lambda} = R(\frac{1}{n_f^2} - \frac{1}{n_i^2})$:$\frac{1}{\lambda_1} = R(\frac{1}{1^2} - \frac{1}{\infty^2}) = R$The series limit of the Balmer series corresponds to the transition from $n = \infty$ to $n = 2$:$\frac{1}{\lambda_3} = R(\frac{1}{2^2} - \frac{1}{\infty^2}) = \frac{R}{4}$The first line of the Lyman series corresponds to the transition from $n = 2$ to $n = 1$:$\frac{1}{\lambda_2} = R(\frac{1}{1^2} - \frac{1}{2^2}) = R(1 - \frac{1}{4}) = R - \frac{R}{4}$Substituting the values of $R$ and $\frac{R}{4}$ from the previous equations:$\frac{1}{\lambda_2} = \frac{1}{\lambda_1} - \frac{1}{\lambda_3}$Rearranging the terms,we get:$\frac{1}{\lambda_1} - \frac{1}{\lambda_2} = \frac{1}{\lambda_3}$

If $\lambda_1$ is the wavelength of the series limit of the Lyman series,$\lambda_2$ is the wavelength of the first line of the Lyman series,and $\lambda_3$ is the series limit of the Balmer series,then the relation between $\lambda_1, \lambda_2,$ and $\lambda_3$ is:

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