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Q.
The wavelengths of $K _{\alpha} X$ -rays for lead isotopes $Pb ^{208}, Pb ^{206}$ and $Pb ^{204}$ are $\lambda_{1}, \lambda_{2}$ and $\lambda_{3}$ respectively.
Then
Wavelengths of the $K_{\alpha}$ lines for given isotopes of lead (Pb) can be given by a general expression
$\frac{1}{\lambda}=R(Z-1)^{2}\left(\frac{1}{1^{2}}-\frac{1}{2^{2}}\right)$
where $R =$ Rydberg's constant, $Z =$ atomic number of the isotopes. Though $P b^{208}, P b^{206}$ and $P b^{204}$ have different atomic masses, $Z$ will be same for them i.e. $82$ .
$\therefore \frac{1}{\lambda_{1}}=R(82-1)^{2}\left(\frac{1}{1^{2}}-\frac{1}{2^{2}}\right)=\frac{3}{4} R(81)^{2}$
$\frac{1}{\lambda_{2}}=\frac{3}{4} R\left(81^{2}\right)$ and $\frac{1}{\lambda_{3}}=\frac{3}{4} R\left(81^{2}\right)$
$\Rightarrow \left(\frac{1}{\lambda_{2}}\right)^{2}=\frac{1}{\lambda_{1}} \times \frac{1}{\lambda_{3}} $
$\Rightarrow \lambda_{2}=\sqrt{\lambda_{1} \lambda_{3}}$