The complex numbers $z_{1}, z_{2}, z_{3}$ are in A.P., then show that they lie on a straight line.

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Solution

Given that $2 z_{2} = z_{1} + z_{3}$
$\Rightarrow \frac{z_{1} - z_{2}}{z_{3} - z_{2}} = - 1$
$\Rightarrow a r g (\frac{z_{1} - z_{2}}{z_{3} - z_{2}}) = a r g (- 1) = 0$
Therefore, $z_{1}$ , $z_{2}$ & $z_{3}$ lie on a straight line.
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