For any integer k- let - k -cos k- 7 -isin k- 7 - where i- -1 - The value of the expression - k-1 12 - k-1 - k - - k-1 3 - 4k-1 - 4k-2 - is

| α_k α_k+1 | represents the side length of a 14 sided polygon (tetradecagon) nbsp;inscribed in a unit circle.Similarly, |α_4k 1 α_4k 2 | repr ...

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Properties of nth Root of a Complex Number

For any integ...

Question

For any integer k, let $α_{k} = cos \frac{k π}{7} + i sin \frac{k π}{7},$ where $i = \sqrt{- 1}$ . The value of the expression $\frac{\sum_{k = 1}^{12} | α_{k + 1} - α_{k} |}{\sum_{k = 1}^{3} | α_{4 k - 1} - α_{4 k - 2} |}$ is

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k

α_{k} = cos (\frac{k π}{7}) + i sin (\frac{k π}{7})

i = \sqrt{- 1}

\frac{12 \sum k = 1 | α_{k + 1} - α_{k} |}{3 \sum k = 1 | α_{4 k - 1} - α_{4 k - 2} |} =

k

a_{k} = cos (\frac{k π}{7}) + i sin (\frac{k π}{7})

\frac{12 \sum k = 1 | a_{k + 1} - a_{k} |}{3 \sum k = 1 | a_{4 k - 1} - a_{4 k - 2} |}

k

a_{k} = cos (\frac{k π}{7}) + i sin (\frac{k π}{7})

\frac{12 \sum k = 1 | a_{k + 1} - a_{k} |}{3 \sum k = 1 | a_{4 k - 1} - a_{4 k - 2} |}

\frac{1}{α_{k} + i} (α_{k} \in R)

8

k = 1, 2, 3, \dots, 8

i = \sqrt{- 1}

\hat{i} + \hat{k} and \hat{i} - \hat{j} + α \vec{k} i s \frac{π}{3}, then α =_____________________.

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