$X$ different wavelengths may be observed in the spectrum from a hydrogen sample if the atoms are excited to states with principal quantum number $n = 6$. The value of $X$ is ..... .

$v_{1}$ is the frequency of the series limit of the Lyman series,$v_{2}$ is the frequency of the first line of the Lyman series,and $v_{3}$ is the frequency of the series limit of the Balmer series. Then:

If the first line of Lyman series has a wavelength $1215.4 \text{ Å}$, the first line of Balmer series is approximately (in $\text{ Å}$)

The ratio of wavelength of spectral lines $H_\alpha$ and $H_\beta$ in the Balmer series is $\frac{x}{20}$. The value of $x$ is $...............$

The ratio of the shortest wavelengths of Bracket and Balmer series of hydrogen atom is

$A$ hydrogen atom emits a photon corresponding to an electron transition from $n=5$ to $n=1$. If $R$ is the Rydberg constant, then the wavelength of the emitted photon is