Question

Let Let $Z_k(k=0,1,2,...............6)$ be the roots of the equation $(z+1{)}^7+z^7=0,$ then ${\sum }_{k=0}^{6}Re\left(zk\right)$ is

• A. 3 - 2i
• B. $0$
• C.
• D. 3 + 2i

Answer 1

The correct option is C

Let $z_k=x_k+i{y}_k,$ we have $(z_k+1{)}^7+z_k^7=0$

$\Rightarrow (z_k+1{)}^7=-z_k^7$

$\Rightarrow \mid z_k+1{\mid }^7+\mid z_k{\mid }^7$

$\Rightarrow \mid z_k+1\mid =\mid z_k\mid$

$\Rightarrow \mid x_k+i{y}_k+1{\mid }^2=\mid x_k+i{y}_k{\mid }^2$

$\Rightarrow (x_k+1{)}^2+y_k^2=x_k^2+y^{2}k$

$\Rightarrow 2x_k+1=0or{x}_k=-\frac{1}{2}$

Thus ${\sum }_{k=0}^{6}Re\left(z_k\right)={\sum }_{k=0}^6{x}_k=-\frac{7}{2}$