b
Lyman Series

Shortest, \(\frac{\mathrm{hc}}{\lambda}=-13.6\left(\frac{1}{\mathrm{n}_1^2}-\frac{1}{\mathrm{n}_2^2}\right)\)

\(\lambda \downarrow \mathrm{E} \uparrow ; \frac{\mathrm{hc}}{\lambda_0}=-13.6(1)\)

Balmer Series:

\(\mathrm{n}=3\)

\(\mathrm{n}=2\)

\(\frac{\mathrm{hc}}{\lambda_1}=-13.6\left(\frac{1}{2^2}-\frac{1}{3^2}\right)\)

\(\frac{\mathrm{hc}}{\lambda_1}=-13.6\left(\frac{1}{4}-\frac{1}{9}\right)\)

\(\frac{\mathrm{hc}}{\lambda_1}=-13.6 \times\left(\frac{5}{36}\right)\)

\(\Rightarrow \frac{-13.6 \lambda_0}{\lambda_1}=-13.6 \times \frac{5}{36}\)

\(\lambda_1=\frac{\lambda_0 \times 36}{5}=\frac{915 \times 36}{5}=6588\)