Question

Let a be a complex number such that |a|<1 and $z_1$,$z_2$,...... be vertices of a polygon such that $z_k$=1+a+$a^2$+.....+$a^{k-1}$. Then the vertices of the polygon lie within a circle

• A.
• B.
• C.
• D.

Answer 1

The correct option is C

We have $z_k$=1+a+$a^2$+.....+$a_{k-1}$ = $\frac{1-a^k}{1-a}$

$\Rightarrow$ $z_k$-$\frac{1}{1-a}$=$\frac{-a^k}{1-a}$

$\Rightarrow$ $|z-\frac{1}{1-a}|$=$\frac{|a{|}^k}{|1-a|}$ < $\frac{1}{|1-a|}$

$\Rightarrow$ $z_k$ lies within $|z-\frac{1}{1-a}|$=$\frac{1}{|1-a|}$