Question

Number of values of z (real or complex) simultaneously satisfying the system of equations

Answer 1

Number of values of x (real or complex) simultaneously satisfying the system of equations 1+z+z2+z3+..........+z17=0 and 1+z+z2+z3+..........+z13=0

Answer 2

Answer:

1

Step-by-step explanation:

We have,

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

Put $1+z+z^{2} ...+z^{13}=0$

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

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

at $z^{14}=0$ ⇒ $z=0$

at $1+z=0$ ⇒ $z=-1$

at $1+z^2=0$ ⇒ $z=i$

⇒ $z=0,-1,i$

But at $z=0,i$

$1+z+z^{2} ...+z^{13}\neq 0$

∴$z=-1$ is the only possible value

Please mark as brainliest...