Discussion Forum

1)

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

   

Answer:

Option C

Explanation:

Given       $\alpha_{k}=\cos \left(\frac{k\pi}{7}\right)+i\sin \left(\frac{k\pi}{7}\right)$

                              $=\cos \left(\frac{2k\pi}{14}\right)+i\sin \left(\frac{2k\pi}{14}\right)$

 $\therefore$    $\alpha_{k}$   are  vertices of regular  polygon having 14 sides.

      Let the side length of regular  polygon be a.

  $\therefore$    $|\alpha_{k+1}-\alpha_{k}|$    = length  ofa side of the regular polygon =a ........(i)

 and  $|\alpha_{4k-1}-\alpha_{4k-2}|$  = length of a side of the regular  polygon=a

$\therefore$          $\frac{\sum_{k=1}^{12}|\alpha_{k+1}-\alpha_{k}|}{\sum_{k=1}^{3}|\alpha_{4k-1}-\alpha_{4k-2}|}=\frac{12(a)}{3(a)}=4$