In the line spectra of hydrogen atom, difference between the largest and the shortest wavelengths of the…

$\frac{1}{{\mathrm{\lambda }}_{\min }}=\mathrm{R}\left(1\right)=\mathrm{R};\mathrm{n}=\infty$ to $1$

$\frac{1}{{\mathrm{\lambda }}_{\max }}=\mathrm{R}\left\{1-\frac{1}{4}\right\}=\frac{3\mathrm{R}}{4}$ ; $\mathrm{n}=2$ to $1$

$\Rightarrow$ ${\mathrm{\lambda }}_{\max }-{\mathrm{\lambda }}_{\min }=\frac{4}{3\mathrm{R}}-\frac{1}{\mathrm{R}}$

$304=\frac{1}{3\mathrm{R}}$                                              ......$\left(\mathrm{a}\right)$

Paschen : ${\mathrm{Y}}_{\mathrm{\lambda }{\text{'}}_{\min }}=\mathrm{R}\left(\frac{1}{9}\right)$

and ${\mathrm{Y}}_{\mathrm{\lambda }{\text{'}}_{\max }}=\mathrm{R}\left(\frac{1}{9}-\frac{1}{16}\right)=\frac{7\mathrm{R}}{16\times 9}$

${{\mathrm{\lambda }}^{\text{'}}}_{\max }-{{\mathrm{\lambda }}^{\text{'}}}_{\min }=\frac{16\times 9}{7\mathrm{R}}-\frac{9}{\mathrm{R}}=\frac{81}{7\mathrm{R}}$       ...$\left(\mathrm{b}\right)$

$\frac{\left(\mathrm{b}\right)}{\left(\mathrm{a}\right)}=\frac{\mathrm{x}}{3\times 304}=\frac{81}{7}$

$\Rightarrow$ $\mathrm{x}=10553.14$