Explanation

(c)
Since, $z_{2} = \frac{z_{1} + z_{3}}{2}\ \ \ \ \ \ \ \ \ \ \ \lbrack\because\ \ \ z_{1},\ z_{2}$ and $z_{3}$ are in AP]
$\Rightarrow \ \ \ \ \ \ B$ is the mid point of the line AC
$\Rightarrow \ \ \ A,\ B,\ C$ are collinear
$\Rightarrow \ \ \ \ \ z_{1},\ z_{2},\ z_{3}$ lie on a straight line