For any integer k , let α k = cos ⁡ k π 7 + i sin ⁡ k π 7 , w h e r e i = - 1 . The value of the expression ∑ k = 1 12 α k + 1 - α k ∑ k = 1 3 α 4 k - 1 - α 4 k - 2 is

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S = ∑ k = 1 12 α k + 1 - α k ∑ k = 1 3 α 4 k - 1 - α 4 k - 2 | α r + 1 - α r | = side length of the polygon having 14 sides = a S = α 2 - α 1 + α 3 - α 2 + α 4 - α 3 + . . . . . . . . . . + α 13 - α 12 α 3 - α 2 + α 7 - α 6 + α 11 - α 10 So, S = 12 a 3 a = 4

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