Complex Numbers 4 Question 19
19. For any integer $k$, let $\alpha _k=\cos \frac{k \pi}{7}+i \sin \frac{k \pi}{7}$, where $i=\sqrt{-1}$. The value of the expression
$$ \frac{\sum _{k=1}^{12}\left|\alpha _{k+1}-\alpha _k\right|}{\sum _{k=1}^{3}\left|\alpha _{4 k-1}-\alpha _{4 k-2}\right|} \text { is } $$
(2016 Adv.)
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Solution:
- Given, $\alpha _k=\cos \frac{k \pi}{7}+i \sin \frac{k \pi}{7}$
$$ =\cos \frac{2 k \pi}{14}+i \sin \frac{2 k \pi}{14} $$
$\therefore \alpha _k$ are vertices of regular polygon having 14 sides.
Let the side length of regular polygon be $a$.
$\therefore\left|\alpha _{k+1}-\alpha _k\right|=$ length of a side of the regular polygon
$$ =a $$
and $\left|\alpha _{4 k-1}-\alpha _{4 k-2}\right|=$ length of a side of the regular polygon
$$ \therefore \quad \frac{\sum _{k=1}^{12}\left|\alpha _{k+1}-\alpha _k\right|}{\sum _{k=1}^{3}\left|\alpha _{4 k-1}-\alpha _{4 k-2}\right|}=\frac{12(a)}{3(a)}=4 $$
