Question

Sa se arate ca (1+2i) la puterea 2014 +(1-2i) la puterea 2014 +i la 24 este nr real ​

Answer 1

$z = (1+2i)^{2014}+(1-2i)^{2014}$

$\begin{aligned}\overline{z} &= \overline{(1+2i)^{2014}}+\overline{(1-2i)^{2014}}\\ &= (\overline{1+2i})^{2014}+(\overline{1-2i})^{2014}\\ &= (1-2i)^{2014}+(1+2i)^{2014}\\ &=(1+2i)^{2014}+(1-2i)^{2014}\\&=z\end{aligned}$

$z=\overline{z} \Rightarrow z\in \mathbb{R}\\ \\ z+i^{24} = z+(i^2)^{12} = z+(-1)^{12} = z+1\in \mathbb{R} \\ \\ \Rightarrow \boxed{(1+2i)^{2014}+(1-2i)^{2014}+i^{24} \in \mathbb{R}}$