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

Question

For any integer k , let &#945; k = cos &#8289; k &#960; 7 + i sin &#8289; k &#960; 7 , &#160; w h e r e &#160; i = &#160; - 1 . &#160; The value of the expression &#8721; k = 1 12 &#945; k + 1 - &#945; k &#8721; k = 1 3 &#945; 4 k - 1 - &#945; 4 k - 2 is

Aicharya · Accepted Answer

S = &#160; &#8721; k = 1 12 &#945; k + 1 - &#945; k &#8721; k = 1 3 &#945; 4 k - 1 - &#945; 4 k - 2 | &#945; r + 1 - &#945; r | &#160; = side length of the polygon having 14 sides = a S = &#160; &#945; 2 - &#945; 1 + &#945; 3 - &#945; 2 + &#945; 4 - &#945; 3 + . . . . . . . . . . + &#945; 13 - &#945; 12 &#945; 3 - &#945; 2 + &#945; 7 - &#945; 6 + &#945; 11 - &#945; 10 So, S = &#160; 12 a 3 a = 4

For any integer k , let α k = cos ⁡ k π 7 + i sin ⁡ k π 7 ,   w h e r e…

Solution

For any integer k , let &#945; k = cos &#8289; k &#960; 7 + i sin &#8289; k &#960; 7 , &#160; w h e r e…

Solution

For any integer k , let α k = cos ⁡ k π 7 + i sin ⁡ k π 7 , w h e r e…