QJEEM22Ma48a6: JEE Main 2022 Mathematics - Class 11 JEEM22Ma48a6 Solution: If , , then is equal to _________.

$\because$ ${z^2} + z + 1 = 0$

$\Rightarrow$ $\omega$ or $\omega$ ²

$\because$ $\left| {\sum\limits_{n = 1}^{15} {{{\left( {{z^n} + {{( - 1)}^n}{1 \over {{z^n}}}} \right)}^2}} } \right|$

$= \left| {\sum\limits_{n = 1}^{15} {{z^{2n}} + \sum\limits_{n = 1}^{15} {{z^{ - 2n}} + 2\,.\,\sum\limits_{n = 1}^{15} {{{( - 1)}^n}} } } } \right|$

$= \left| {0 + 0 - 2} \right|$

$= 2$