Question

If $|z_1|=|z_2|=....=|z_n|=1$, then $|\frac{1}{z_1}+\frac{1}{z_2}+......+\frac{1}{z_n}|$ is

• A. equal to $|z_1+z_2+......+z_n|$
• B. less than $|z_1+z_2+......+z_n|$
• C. greater than $|z_1+z_2+......+z_n|$
• D. equal to 1

Answer 1

The correct option is A equal to $|z_1+z_2+......+z_n|$

Given, $|z_1|=|z_2|=......=|z_n|=1$ ........ (i)
Now, $|\frac{1}{z_1}+\frac{1}{z_2}+.....+\frac{1}{z_n}|=|\frac{{\overline{z}}_1}{z_1{\overline{z}}_1}+\frac{{\overline{z}}_2}{z_2{\overline{z}}_2}+.....\frac{{\overline{z}}_n}{z_n{\overline{z}}_n}|$
$=|\frac{{\overline{z}}_1}{|z_1{|}^2}+\frac{{\overline{z}}_2}{|z_2{|}^2}+......\frac{{\overline{z}}_n}{|z_n{|}^2}|=|{\overline{z}}_1+{\overline{z}}_2+.......+{\overline{z}}_n|$ from (i)
$=|z_1+z_2+.........+z_n|(\because |\overline{z}|=|z|)$