The largest wavelength in the ultraviolet region of the hydrogen spectrum is 122nm. The smallest wavelength in the infrared region of the hydrogen spectrum (to the nearest integer) is:

Largest wavelength in the ultraviolet region of the hydrogen spectrum corresponds to the transition from  $n=2$  to  $n=1,$
$$\begin{gathered} \frac{1}{\lambda }=R_z^2(\frac{1}{n_1^2}-\frac{1}{n_2^2}) \\ \Rightarrow \frac{1}{122}=R(\frac{1}{1^2}-\frac{1}{2^2})=R(1-\frac{1}{4}) \\ \Rightarrow R=\frac{1}{122}\times \frac{4}{3} \end{gathered}$$ 
Transition from  $n=\infty ton=3$ will produce the smallest wavelength in infrared region of hydrogen spectrum so
 $$\begin{gathered} \frac{1}{\lambda }=R(\frac{1}{3^2}-\frac{1}{\infty }) \\ \Rightarrow \frac{1}{\lambda }=\frac{1}{122}\times \frac{4}{3}\times \frac{1}{9} \\ \Rightarrow \frac{1}{\lambda }=\frac{122\times 3\times 9}{4}=823.5nm \end{gathered}$$