If the radius of the $3^{\text{rd}}$ Bohr's orbit of hydrogen atom is $r_{3}$ and the radius of $4^{\text{th}}$ Bohr's orbit is $r_{4}$,then which of the following relations is correct?

The total energy of the electron of $H$ atom in the second quantum state is $-E_2$. The total energy of the $He^{+}$ ion in the third quantum state is

The potential energy of an electron in the third excited state of $He^{+}$ ion is ............. $eV$.

Assertion : The radius of the first orbit of hydrogen atom is $0.529 \ \mathring{A}$.
Reason : Radius of each circular orbit $(r_n) = 0.529 \ \mathring{A} \ (n^2/Z)$,where $n = 1, 2, 3$ and $Z =$ atomic number.

The work function of a metal is $4.2 \, eV$. If radiation of $2000 \, \mathring{A}$ falls on the metal,then the kinetic energy of the fastest photoelectron is:

When an electric discharge is passed through hydrogen gas,the hydrogen molecules dissociate to produce excited hydrogen atoms. These excited atoms emit electromagnetic radiation of discrete frequencies which can be given by the general formula
$\bar{v} = 109677 \left[ \frac{1}{n_{i}^{2}} - \frac{1}{n_{f}^{2}} \right] \ cm^{-1}$
What points of Bohr's model of an atom can be used to arrive at this formula? Based on these points,derive the above formula,giving a description of each step and each term.