# How find the maximum of the $|z_{1}+z_{2}+\cdots+z_{n}|$

Let $n$ be give postive integers,$z_{i}(i=1,2,\cdots,n)$ be postive integers,and such $|Re(z_{i})|+|Im(z_{i})|=3$,find the maximum of the $$|z_{1}+z_{2}+\cdots+z_{n}|$$

I try use $$|z_{1}+z_{2}+\cdots+z_{n}|\le\sum_{i=1}^{n}|z_{i}|$$

• Did you really mean to write that the $z_i$ are positive integers? – user940 Aug 6 '17 at 16:06

continuing your inequality$$|z_{ 1 }+z_{ 2 }+\cdots +z_{ n }|\le \sum _{ i=1 }^{ n } |z_{ i }|\le \sum _{ i=1 }^{ n } Re\left| { z }_{ i } \right| +\sum _{ i=1 }^{ n } Im|z_{ i }|=3n$$
• when the inequality $=3n$ – wightahtl Aug 6 '17 at 15:31
• @wightahtl,$|z_{ 1 }+z_{ 2 }+\cdots +z_{ n }|\le \sum _{ i=1 }^{ n } |z_{ i }|=\sum _{ i=1 }^{ n } \left| Re\left( { z }_{ i } \right) +iIm\left( { z }_{ i } \right) \right| \le \sum _{ i=1 }^{ n } Re|z_{ i }|+\sum _{ i=1 }^{ n } Im|z_{ i }|=\\ =\underset { n }{ \underbrace { \left( { a }_{ 1 }+{ b }_{ 1 } \right) +\left( { a }_{ 2 }+{ b }_{ 2 } \right) +...+\left( { a }_{ n }+{ b }_{ n } \right) } } =3n$ – haqnatural Aug 6 '17 at 15:36