Question

Number of values of $z$ (real or complex) simultaneously satisfying the system of equations $1+z+z^2+z^3+.....+z^{17}=0$ and $1+z+z^2+z^3+.....+z^{13}=0$ is

• A. $1$
• B. $2$
• C. $3$
• D. $4$

Answer 1

The correct option is A $1$

We have,

$1+z+z^2\cdots +z^{17}=(1+z+z^2\cdots +z^{13})+z^{14}(1+z+z^2+z^3)=0$

Put $1+z+\cdots +z^{13}=0$

$z^{14}(1+z+z^2+z^3)=0$

$\Rightarrow z^{14}(1+z)(1+z^2)=0$

$\Rightarrow z=0,i,-1$

But at $z=0,i,1+z+z^2+\cdots +z^{13}\ne 0$

$\therefore z=-1$ is the only possible value.