_ k=1 ^6 ( 2 k 7 -i 2 k 7 )=

Step-by-step solution

aligned & _ k=1 ^6 ( 2 k 7 -i 2 k 7 ) & =-i _ k=1 ^6 [ ( 2 k 7 )- 1 i ( 2 k 7 ) ] & =-i _ k=1 ^6 [ ( 2 k 7 )+i ( 2 k 7 ) ] & =-i [ _ k=1 ^6 e^ i 2 k 7 ] (i) e^ i = +i aligned array r Now, [1+e^ i 2 7 +e^ i 4 7 +e^ i 6 7 +e^ i 8 7 +e^ i 10 7 +e^ i 12 7 =0 . z^7=1 then, roots 1, e^ i 2 7 , e^ i 4 7 e^ i 12 7 =1+ _ k=1 ^6 e^ i(2 k / 7) =0= _ k=1 ^6 e^ i(2 k / 7) =-1 (ii) array From Eqs. (i) and (ii), we get =-i(-1)=i