$\frac{{Maximum\,\,\lambda \,\,of\,\,Lyman\,\,series}}{{Maximum\,\,\lambda \,\,of\,\,Balmer\,\,series}}\,\, = \,\,\frac{{{1^{st}}\,line\,\,of\,\,Lyman\,\,series}}{{{1^{st}}\,line\,\,of\,\,Balmer\,\,series}}$

$\frac{{Lyman}}{{Balmer}}\,\, = \,\,\frac{{{{\frac{1}{\lambda }}_L}}}{{{{\frac{1}{\lambda }}_B}}}\,\,\, \Rightarrow \,\,\,\,\,\,\,\frac{{{{\frac{1}{\lambda }}_L}}}{{{{\frac{1}{\lambda }}_B}}}\,\, = \,\,\frac{{R\left[ {\frac{1}{{{1^2}}} - \frac{1}{{{2^2}}}} \right]}}{{R\left[ {\frac{1}{{{2^2}}} - \frac{1}{{{3^2}}}} \right]}}\,\, = \,\,\frac{{R\left[ {\frac{1}{1} - \frac{1}{4}} \right]}}{{R\left[ {\frac{1}{4} - \frac{1}{9}} \right]}}\,\,\, \Rightarrow \,\frac{{R\left[ {\frac{3}{4}} \right]}}{{R\left[ {\frac{5}{{36}}} \right]}}\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\frac{{{{\frac{1}{\lambda }}_L}}}{{{{\frac{1}{\lambda }}_B}}}\,\, = \,\,\frac{{27}}{5} \Rightarrow \frac{{{\lambda _L}}}{{{\lambda _B}}}\,\, = \,\,\frac{5}{{27}}$