For any integer k, let `alpha _h= cos. (k pi)/(7)+ I sin. (k pi)/(7) " where if "=sqrt(-1).` The value of the expression
`(Sigma_(k=1)^(12) |alpha_(k+1)-alpha_k|)/(Sigma_(k=1)^(3) |alpha_(4k+1)-alpha_(4k-2)|)`is

Answer

4

Explanation

Given `alpha_(k) = cos ((kpi)/(7)) + i sin ((k pi)/(7))`
`= cos ((2k pi)/(14)) + i sin ((2 kpi)/(14))`
`therefore alpha_(k)` are vertices of regular polygon having 14 sides .
Let the side length of regular polygon be `alpha`.
`therefore |alpha_(k + 1) - alpha_(k)| ` = length of a side of the regular polygon
`= alpha " " ... (i)`
and `|alpha_(4k-1) - alpha_(4k-2)|` = length of a side of the regular polygon
` = alpha " " .... (ii)`
`therefore ( sum_(h=1)^(12) |alpha_(k+1) - alpha_(k)|)/(sum_(h=1)^(3) |alpha_(4k-1) - alpha_(4k-2)|) = (12(a))/(3(a)) = 4`