MCQ
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

$\frac{\sum_{k=1}^{12}\left|\alpha_{k+1}-\alpha_k\right|}{\sum_{k=1}^3\left|\alpha_{4 k-1}-\alpha_{4 k-2}\right|}$ is

  • A
    $1$
  • B
    $2$
  • C
    $3$
  • $4$

Answer

Correct option: D.
$4$
d
$\frac{\sum_{k=1}^{12}\left|e^{i \frac{k x}{7}}\right|\left|e^{i \frac{\pi}{7}}-1\right|}{\sum_{k=1}^3\left|e^{i(4 k-2)}\right|\left|e^{i \frac{\pi}{7}}-1\right|}=\frac{12}{3}=4$

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