Algebra Level 3

$\dfrac{\displaystyle \sum_{k=1}^{12} |\alpha_{k+1}-\alpha_k|}{\displaystyle \sum_{k=1}^3 |\alpha_{4k-1}-\alpha_{4k-2}|}$

For integer $$k$$, let $$\alpha_k=\cos \left( \frac{k \pi}{7} \right)+i \sin \left( \frac{k \pi}{7} \right)$$, where $$i=\sqrt{-1}$$. Evaluate the value of the expression above.

×