Question

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

• A. $0$
• B. $\frac{3}{2}$
• C. $-\frac{7}{2}$
• D. $\frac{7}{2}$

Answer 1

The correct option is C $-\frac{7}{2}$

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

$\Rightarrow {(z_k+1)}^7=-{z_k}^7\Rightarrow {|z_k+1|}^7={\left|z_k\right|}^7\Rightarrow |z_k+1|=\left|z_k\right|\Rightarrow {|x_k+i{y}_k+1|}^2={|x_k+i{y}_k|}^2\Rightarrow {(x_k+1)}^2+{y_k}^2={x_k}^2+{y_k}^2$

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

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