c
Shortest wavelength in Balmer series when transition of \(e^{-}\)from \(\infty\) to \(n=2\)

\(\because \frac{1}{\lambda}= Rz ^2\left[\frac{1}{2^2}-\frac{1}{\infty^2}\right]\)

\(\frac{1}{\lambda}=\frac{ R }{4}\)

Shortest wavelength is Bracket series when transition of \(e^{-}\)from \(\infty\) to \(n=4\)

\(\frac{1}{\lambda^{\prime}}= R (1)^2\left[\frac{1}{4^2}-\frac{1}{\infty^2}\right] \Rightarrow \frac{1}{\lambda^{\prime}}=\frac{ R }{16}\)

Eq. \((1)\)/Eq.\((2)\)

\(\frac{\lambda^{\prime}}{\lambda}=\frac{R}{4} \times \frac{16}{R} \Rightarrow \lambda^{\prime}=4 \lambda\)