Question

Let $|z_r-r|\le r$, for all $r=1,2,3,...,n.$ Then $\left|{\sum }_{r=1}^n{z}_r\right|$ is less than

• A. $n$
• B. $2n$
• C. $n(n+1)$
• D. $\frac{n(n+1)}{2}$

Answer 1

The correct option is C $n(n+1)$

$|z_1-1|\le 1,|z_2-2|\le 2,|z_3-3|\le \mathrm{3...}|z_n-n|\le n$
Adding these and using triangle inequality:
$|z_1+z_2+...z_n-(1+2+...n\left)\right|\le 1+2+...n=>|z_1+z_2+...z_n-\left(\frac{n(n+1)}{2}\right)|\le \frac{n(n+1)}{2}$
Thus, $|z_1+z_2+...z_n|-\left(\frac{n(n+1)}{2}\right)\le \frac{n(n+1)}{2}=>|z_1+z_2+...z_n|\le n(n+1)$
Hence, (c) is correct.