Question

If three complex numbers are in AP then they lie on

Please provide the answer with explanation for how the concept of AP is used here.

Answer 1

let three numbers be a+id ,b+id,c+id
they will be pointed as (a,d) (b,d) (c,d) in plane

if they are in AP,
2(b+id) = a+id + c+id
= a+c +2(id)
2b+i(2d) = a+c +i(2d)

so 2b = a + c.

comparing to the points, all the points have same y coordinate,
so 2b= a+c implies (b,d) is the midpoint of the line joining the points (a,d) and (c,d)

that means the three complex numbers lies on a straight line in complex plane.

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