For any integer  $k,$${\alpha }_k=\cos \left(\frac{k\pi }{7}\right)+i\sin \left(\frac{k\pi }{7}\right)$ ,where $i=\sqrt{-1}$  . The value of expression $\frac{\sum _{k=1}^{12} \left|{\alpha }_{k+1}-{\alpha }_k\right|}{\sum _{k=1}^3 \left|{\alpha }_{4k-1}-{\alpha }_{4k-2}\right|}$ is 

 

${\alpha }_k=\cos \frac{2k\pi }{14}+i\sin \frac{2k\pi }{14}=e^{i\frac{2\pi k}{14}}$

$\frac{\sum _{k=1}^{12} \left|e^{i2(k+1)\frac{\pi }{14}}-e^{i\frac{2k\pi }{14}}\right|}{\sum _{k=1}^3 \left|e^{\frac{i2(4k-1)\pi }{14}}-e^{i\frac{2(4k-2)\pi }{14}}\right|}=\frac{\sum _{k=1}^{12} \left|e^{i\frac{2\pi }{14}}-1\right|}{\sum _{k=1}^3 \left|e^{i\frac{2\pi }{14}-1}\right|}=\frac{12}{3}=4$

The expression has a geometric meaning.$\left|{\alpha }_k+1-{\alpha }_k\right|$ is the side length of a polygon having 14 sides as ${\alpha }_k^{\mathrm{\prime }}s$ are 14 ^th roots of unity.

The expression $=\frac{12}{3}=4$

Recall, ${\alpha }_k=\cos \frac{k\pi }{7}+i\sin \frac{k\pi }{7}=\cos \frac{2\pi k}{14}+i\sin \frac{2k\pi }{14}$