Finding orbit radius using the Bohr model and Rydberg equation
To answer the second part:
We know $M=208m_e$, $Z=3$, $\hbar=\frac{h}{2\pi}$.
Part one has a mistake, as it is
$$
\begin{align}
&amp;&amp;\frac{Mv^2}{r}&amp;=\mathcal{k_e}\cdot\frac{(\mathcal{e})(Z\mathcal{e})}{r^2}\\
&amp;&amp;Mvr&amp;=n\hbar\\
\implies&amp;&amp; r &amp;=\frac{n^2\hbar^2}{M\cdot\mathcal{k_e}\cdot Z \mathcal{e}^2}
\end{align}
$$
We also know the Bohr radius:
$$a_0= \mathcal{k_e}\cdot\frac{\hbar^2}{m_e\cdot\mathcal{e}^2} \approx 5{,}29 \cdot 10^{-11}\mathrm{m}$$
Therefore we can write and cancel:
$$
\begin{align}
&amp;&amp; r&amp;=a_0\\
&amp;&amp; \frac{\color{\green}{\hbar^2}}{\color{\red}{\mathcal{k_e}}\cdot m_e\cdot\color{\navy}{\mathcal{e}^2}} &amp;=\frac{n^2\color{\green}{\hbar^2}}{M\cdot\color{\red}{\mathcal{k_e}}\cdot Z \color{\navy}{\mathcal{e}^2}}\\
\therefore&amp;&amp; Z\frac{M}{m_e}&amp;=n^2\\
\therefore&amp;&amp; n&amp;=\sqrt{Z\cdot208}\approx25
\end{align}
$$
The third part:
The Rydberg Formula is given as
$$\frac{1}{\lambda_{\mathrm{vac}}} = \mathcal{R}Z^2 \left(\frac{1}{n_1^2}-\frac{1}{n_2^2}\right)$$
with the Rydberg $\mathcal{R}$ constant defined for a photon emitted by an electron. We'll assume that the mass of the nucleus is 7 atomic units (three protons + four neutrons). Taking into account that $m_p\approx 1836m_e$, we arrive at
$$\mathcal{R} =\frac{\mathcal{R}_\infty}{1+\frac{M}{T}} = \frac{\mathcal{R}_\infty}{1+\frac{208m_e}{7\cdot1836m_e}}$$
Now the Rydberg constant has to be modified to include the mass of the particle:
$$\mathcal{R}_\infty = \frac{M e^4}{8 c \varepsilon_0^2 h^3}=208\mathcal{R}_{m_e,\infty}$$
With $\mathcal{R}_{m_e,\infty} = 1.097\cdot 10^7~\mathrm{m^{-1}}$ (wikipedia), I got to $\lambda_\mathrm{vac}=55.6~\mathrm{pm}$.
Without taking the reduced mass into account, i.e. $\mathcal{R}\approx\mathcal{R}_\infty$ I arrived at $\lambda_\mathrm{vac}=54.8~\mathrm{pm}$.
Both values are reasonably close to the solution given.
(If the question was really about the muon, the more accurate weight ratio is 206.77 and the corresponding wavelengths 55.1 pm and 56.0 pm.)