If  $z_1=\cos \theta +i\sin \theta and1,z_1,z_1^2,z_1^3,\mathrm{........},{z_1}^{n-1}$ are vertices  of a  regular  polygon  such  that $\frac{I_m\left(z_1^2\right)}{R_e\left(z_1\right)}=\frac{\sqrt{5}-1}{2},$ then the sum of  the digits of n is 

$z_1=\cos \theta +i\sin \theta and1,z_1,z_1^2,z_1^3,\mathrm{.............}{z_1}^{n-1}$   are  vertices  of  a regular  polygon 
 $$\begin{gathered} \theta =\frac{2\pi }{n},z_1=\cos \frac{2\pi }{n}+i\sin \frac{2\pi }{n},z_1^2=e^{i4\frac{\pi }{4}}+i\sin \frac{4\pi }{n} \\ \frac{lm\left(z_1^2\right)}{\mathrm{Re}\left(z_1\right)}=\frac{\sqrt{5}-1}{2} \\ \Rightarrow \frac{\sin \frac{4\pi }{n}}{\cos \left(\frac{2\pi }{n}\right)}=\frac{\sqrt{5}-1}{2}\Rightarrow \frac{2\sin \left(\frac{2\pi }{n}\right)\cos \left(\frac{2\pi }{n}\right)}{\cos \left(\frac{2\pi }{n}\right)}=\frac{\sqrt{5}-1}{2} \\ \Rightarrow \sin \left(\frac{2\pi }{n}\right)=\frac{\sqrt{5}-1}{4}=\sin {18}^0=\sin \frac{\pi }{10} \Rightarrow \frac{2\pi }{n}=\frac{\pi }{10} \\ \Rightarrow n=20 \end{gathered}$$