Question

If $Re\left(z\right)$ is a positive integer, then value of the $|1+z+...+z^n|$ cannot be less than

• A. $|z^n|-\frac{1}{|z|}$
• B. $|z^n|+\frac{1}{|z|}$
• C. $n|z{|}^n$
• D. $n|z{|}^n+1$

Answer 1

Answer: (A)

$|z^n|-\frac{1}{|z|}$

$|1+z+z^2+..z^n|\le 1+|z|+|z{|}^2+..|z{|}^n$
$\le \frac{|z{|}^{n+1}-1}{|z|-1}$
$\le \frac{(|z|.|z{|}^n-1)}{|z|-1}$
$\le \frac{|z|}{|z|-1}.[|z{|}^n-\frac{1}{|z|}]$
Hence
It cannot be less than $[|z{|}^n-\frac{1}{|z|}]$.