For any integer `k ,` let `alpha_k=cos(kpi)/7+isin(kpi)/7,` where `i=sqrt(-1)dot` Value of the expression `(sum_(k=1)^(12)|alpha_(k+1)-alpha_k|)/(sum_(k=1)^(13)|alpha_(4k-1)-alpha_(4k-2)|)` is

Options

A. 8
B. 6
C. 4
D. 2

Answer

C

Explanation

We have, `alpha_(k) = cos(kpi)/7 + isin(kpi)/7=e^(ikpi)/7`
`therefore |alpha_(k)|=1` and `alpha_(k+1) = alpha_(1)alpha_(k),alpha_(4k-1)=alpha_(4k-2)alpha_(1)`
Now, `sum_(k=1)^(12)|alpha_(k+1)-alpha_(k)|=sum_(k=1)^(12)|alpha_(k)||alpha_(1)-1|=sum_(k=1)^(12)(alpha_(1)-1)=12(alpha_(1)-1)`
and `sum_(k=1)^(3) |alpha_(4k-1)-alpha_(4k-2)|=sum_(k=1)^(3)|alpha_(4k-2)||alpha_(1)-1|=sum_(k=1)^(3) |alpha_(1)-1|=3(alpha_(1)-1)`
`(sum_(k=1)^(12)|alpha_(k+1)-alpha_(k)|)/(alpha_(k=1)^(3) |alpha_(4k-1)-alpha_(4k-2)|)=12((alpha_(1)-1)/(3(alpha_(1)-1)))=4`