According to Bohr's model, the radius of the second orbit of a helium ion $(He^+)$ is ........ $\mathring{A}$.

If the difference between $(n + 1)^{th}$ Bohr radius and $n^{th}$ Bohr radius is equal to the $(n - 1)^{th}$ Bohr radius, then the value of $n$ is:

The Rydberg constant $R$ for hydrogen is

In a hydrogen atom,the radius of the $n^{th}$ Bohr orbit is $r_n$. The graph between $\log \left( \frac{r_n}{r_1} \right)$ and $\log n$ will be:

$A$ particle of mass $m$ is moving in a circular orbit under the influence of the central force $F(r) = -kr$,corresponding to the potential energy $V(r) = \frac{1}{2}kr^2$,where $k$ is a positive force constant and $r$ is the radial distance from the origin. According to Bohr's quantization rule,the angular momentum of the particle is given by $L = n\hbar$,where $\hbar = \frac{h}{2\pi}$,$h$ is Planck's constant,and $n$ is a positive integer. If $v$ and $E$ are the speed and total energy of the particle,respectively,then which of the following expression$(s)$ is(are) correct?
$(A)$ $r^2 = n\hbar \sqrt{\frac{1}{mk}}$
$(B)$ $v^2 = n\hbar \sqrt{\frac{k}{m^3}}$
$(C)$ $\frac{L}{mr^2} = \sqrt{\frac{k}{m}}$
$(D)$ $E = \frac{n\hbar}{2} \sqrt{\frac{k}{m}}$

The energy equivalent to the Rydberg constant is ...... $eV$.