For any integer k,let w_k = cos left( frac{kp | vedclass

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(B) Given $w_k = e^{i \frac{k\pi}{11}}$.Note that $|w_{m} - w_{n}| = |e^{i \frac{m\pi}{11}} - e^{i \frac{n\pi}{11}}| = |e^{i \frac{n\pi}{11}} (e^{i \f

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$2$

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(B) Given $w_k = e^{i \frac{k\pi}{11}}$.Note that $|w_{m} - w_{n}| = |e^{i \frac{m\pi}{11}} - e^{i \frac{n\pi}{11}}| = |e^{i \frac{n\pi}{11}} (e^{i \frac{(m-n)\pi}{11}} - 1)| = |e^{i \frac{(m-n)\pi}{11}} - 1| = |\cos \frac{(m-n)\pi}{11} + i \sin \frac{(m-n)\pi}{11} - 1| = \sqrt{(\cos \frac{(m-n)\pi}{11} - 1)^2 + \sin^2 \frac{(m-n)\pi}{11}} = \sqrt{2 - 2 \cos \frac{(m-n)\pi}{11}} = 2 |\sin \frac{(m-n)\pi}{22}|$.For the numerator: $\sum_{k=1}^8 |w_{2k+1} - w_{2k}| = \sum_{k=1}^8 |e^{i \frac{(2k+1)\pi}{11}} - e^{i \frac{2k\pi}{11}}| = \sum_{k=1}^8 |e^{i \frac{2k\pi}{11}} (e^{i \frac{\pi}{11}} - 1)| = \sum_{k=1}^8 |e^{i \frac{\pi}{11}} - 1| = 8 |e^{i \frac{\pi}{11}} - 1|$.For the denominator: $\sum_{k=1}^4 |w_{3k-1} - w_{3k-2}| = \sum_{k=1}^4 |e^{i \frac{(3k-1)\pi}{11}} - e^{i \frac{(3k-2)\pi}{11}}| = \sum_{k=1}^4 |e^{i \frac{(3k-2)\pi}{11}} (e^{i \frac{\pi}{11}} - 1)| = \sum_{k=1}^4 |e^{i \frac{\pi}{11}} - 1| = 4 |e^{i \frac{\pi}{11}} - 1|$.Therefore,the ratio is $\frac{8 |e^{i \frac{\pi}{11}} - 1|}{4 |e^{i \frac{\pi}{11}} - 1|} = \frac{8}{4} = 2$.

For any integer $k$,let $w_k = \cos \left( \frac{k\pi}{11} \right) + i \sin \left( \frac{k\pi}{11} \right)$,where $i = \sqrt{-1}$. The value of the expression $\frac{\sum_{k=1}^8 |w_{2k+1} - w_{2k}|}{\sum_{k=1}^4 |w_{3k-1} - w_{3k-2}|}$ is

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