Question

Match the entries of List A and List B

Answer 1

$A].arg\frac{z_1}{z_2}=arg(z_1)-arg(z_2)=arg(w)-arg(1)=\frac{2\pi }{3}$

$B].$ if $w$ is $n^{th}$ root of unity, then;

$1+w+w^2+w^3.......+w^{n-1}=0⟹w+w^2+.....+w^{n-1}=-1$

$C].$ Applying twice your method of multiplying by $w$ and subtracting, and using $w^n=1$

$1+2w+3w^2+\cdots +n{w}^{n-1}=x$

$w+2w^2+3w^3+\cdots +n{w}^n=xw$

$1+w+w^2+\dots w^{n-1}=x(1-w)+n$

$w+w^2+w^3+\cdots +w^n=x(1-w)w+nw$

$1-1=x(1-w)w+nw$

$x=\frac{-n}{1-w}$

$D].$ $(1-w)(1-w^2)......(1-w^{n-1})=n$

$E]$. In general $w^3=1$. Hence

$w_1^4{w}_2^4=w_1{w}_2$