As the quantum number $n$ increases,the difference in energy between consecutive energy levels

An electron jumps from the $4th$ orbit to the $2nd$ orbit of a hydrogen atom. Given the Rydberg constant $R = 10^5 \text{ cm}^{-1}$. The frequency in $\text{Hz}$ of the emitted radiation will be:

The number of spectral lines emitted by atomic hydrogen that is in the $4^{\text{th}}$ energy level is

Energy of the electron in $n^{th}$ orbit of hydrogen atom is given by $E_n = - \frac{13.6}{n^2} \ eV$. The amount of energy needed to transfer an electron from the first orbit to the third orbit is........$eV$.

$A$ sample of hydrogen atoms is in an excited state (all the atoms). The photons emitted from this sample are made to pass through a filter through which light having a wavelength greater than $800 \ nm$ can only pass. Only one type of photon is found to pass through the filter. The sample's initial excited state is: [Take $hc = 1240 \ eV \cdot nm$,ground state energy of hydrogen atom = $-13.6 \ eV$.]

Which of the following transitions in hydrogen atoms emit photons of highest frequency?