Answer
See the detailed answer below.
Work Step by Step
$$\color{blue}{\bf [a]}$$
For the Balmer series, we need to use the Balmer formula for the hydrogen atom,
$$\lambda=\dfrac{91.18\;\rm nm}{\dfrac{1}{m^2}-\dfrac{1}{n^2}}$$
where $m=1,2,3,...$, and $n=m+1,m+2,m+3,...$
For the first wavelength 102.6 nm,
$$ \dfrac{1}{m^2}-\dfrac{1}{n^2}=\dfrac{91.18\;\rm nm}{\lambda_1}$$
$$ \dfrac{n^2-m^2}{m^2n^2}=\dfrac{91.18 }{102.6 }=0.8887$$
This works for $m=\color{red}{\bf 1}$, and $n= \color{red}{\bf 3}$
For the second wavelength 1876 nm,
$$ \dfrac{1}{m^2}-\dfrac{1}{n^2}=\dfrac{91.18\;\rm nm}{\lambda_2}$$
$$ \dfrac{n^2-m^2}{m^2n^2}=\dfrac{91.18 }{1876 }=0.0486$$
This works for $m=\color{red}{\bf 3}$, and $n =\color{red}{\bf 4}$
$$\color{blue}{\bf [b]}$$
We know that the visible light is in the range of
$$400\;{\rm nm}\lt \lambda_{\rm visible}\lt 700\;{\rm nm}$$
where 400 nm and less is ultraviolet and 700 nm and more is infrared.
This means that the 102.6 nm is in the ultraviolet region while the 1876 nm is in the infrared region.
