Mathematics

Problem

Let

a

be a complex number such that

|a| < 1

and

z_1, z_2, \dots, z_n

be the vertices of a polygon such that

z_k = 1 + a + a^2 + \dots + a^k

, then show that vertices of the polygon lie within the circle

\left|z - \frac{1}{1-a}\right| = \frac{1}{|1-a|}