Answer
a) $\rm Dark$
b) $1.597$
Work Step by Step
$$\color{blue}{\bf [a]}$$
We know that the arms of the interferometer are equal. So if there is no crystal, the output will be a bright fringe [since we have the same light wavelength and the same distance traveled by this wavelength].
So to find if the output, with the crystal, is bright or dark, we need to find the number of wavelengths inside the crystal and compare them to the number of wavelengths at the same distance outside the crystal.
If both are an integer number then the output is bright if not then it varies.
The number of wavelengths inside the crystal is given by
$$N_{\rm inside}=\dfrac{L_{\rm crystal }}{\lambda_{\rm crystal }}$$
where $\lambda_{\rm crystal }=\lambda/n_{\rm crystal }$
$$N_{\rm inside}=\dfrac{L_{\rm crystal }n_{\rm crystal }}{\lambda }\tag 1$$
The number of wavelengths outside the crystal is given by
$$N_{\rm outside}=\dfrac{L_{\rm crystal } }{\lambda }\tag 2$$
Divide (1) by (2);
$$\dfrac{N_{\rm inside}}{N_{\rm outside}}=\dfrac{\dfrac{L_{\rm crystal }n_{\rm crystal }}{\lambda }}{\dfrac{L_{\rm crystal } }{\lambda }}=n_{\rm crystal }$$
where $n_{\rm crystal }$ here is the index of refraction of the crystal without applying voltage.
$$\dfrac{N_{\rm inside}}{N_{\rm outside}}=1.522 $$
Therefore,
$$N_{\rm inside}=\color{red}{\bf1.5 }N_{\rm outside}$$
And since the number of wavelengths inside the crsytal is one and half the number of wavelengths outside it, then the output must be $\rm \bf dark$.
$$\color{blue}{\bf [b]}$$
Now we need to find the index of refraction of the crystal with applying voltage that makes the output is bright.
Plugging the known into (1) to find the number of wavelengths inside the crystal before applying the voltage.
$$N_{\rm inside}=\dfrac{(6.70\times 10^{-6}) (1.522)}{(1\times 10^{-6})} =\bf 10.2$$
Plugging the known into (2) to find the number of wavelengths,
$$N_{\rm outside}=\dfrac{(6.70\times 10^{-6}) }{(1\times 10^{-6})} =\bf 6.7$$
We need to add one-half wavlength inside the crystal.
So the number inside the crystal had to be 10.7 so we got an integer number of wavelengths as out side.
Plugging 10.7 into (1) and solving for $n_{\rm crystal}$
$$N_{\rm inside}=10.7=\dfrac{(6.70\times 10^{-6}) n_{\rm crystal}}{(1\times 10^{-6})} $$
$$n_{\rm crystal}=\dfrac{10.7(1\times 10^{-6})}{(6.70\times 10^{-6})}=\color{red}{\bf 1.597}$$
