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

For any integer k, let αₖ=cos(kπ/7)+isin(kπ/7), where i=√-1. The value of the expression (∑ₖ₌₁¹²|αₖ₊₁-αₖ|)/(∑ₖ₌₁³|α₄ₖ₋₁-α₄ₖ₋₂|) is

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

$i=\sqrt{-1},\quad k\in\mathbb{Z}$

$\text{To find: } \dfrac{\sum_{k=1}^{12}\left|\alpha_{k+1}-\alpha_{k}\right|}{\sum_{k=1}^{3}\left|\alpha_{4k-1}-\alpha_{4k-2}\right|}$

$\alpha_k = e^{i k\pi/7}$

$\text{Argand plane: } z = x + iy$

$|\alpha_k|^2 = \cos^2\!\left(\frac{k\pi}{7}\right)+\sin^2\!\left(\frac{k\pi}{7}\right) = 1$

$\Rightarrow\ |\alpha_k| = 1$

$\alpha_{k+7} = e^{ik\pi/7}\,e^{i\pi} = -\alpha_k$

$1.\ \text{show } |\alpha_{k+1}-\alpha_k| \text{ is constant}$

$2.\ \text{write the sums using it}$

$3.\ \text{take the ratio}$

$|\alpha_2 - \alpha_1| = ?$

OA = OB = 1, ∠ AOB = π/7

$\text{base} = 2\sin\frac{\theta}{2}, \quad \theta = \frac{\pi}{7}$

$|\alpha_{k+1}-\alpha_k| = 2\sin\frac{\pi}{14} \quad \text{for every } k$

$|\alpha_{k+1}-\alpha_k| = |e^{ik\pi/7}|\,|e^{i\pi/7}-1|$

$= |e^{i\pi/7}-1| = 2\sin\frac{\pi}{14}$

$\sum_{k=1}^{12}|\alpha_{k+1}-\alpha_k| = 12\cdot 2\sin\frac{\pi}{14}$

$k=1:\ |\alpha_3-\alpha_2|,\quad k=2:\ |\alpha_7-\alpha_6|$

$k=3:\ |\alpha_{11}-\alpha_{10}|$

$\sum_{k=1}^{3}|\alpha_{4k-1}-\alpha_{4k-2}| = 3\cdot 2\sin\frac{\pi}{14}$

$\dfrac{12\cdot 2\sin\frac{\pi}{14}}{3\cdot 2\sin\frac{\pi}{14}}$

$= \dfrac{12}{3} = 4$

$\boxed{4}$