Question

aratati ca (1+i)^n + (1-i)^n apartine nr reale

Answer 1

$Un \ numar \ complex \ z \ este \ real \ daca \ z=\overline{z}\\ \\ z=(1+i)^n + (1-i)^n\\ \\ \overline{z}=\overline{(1+i)^n + (1-i)^n}=\\ \\ =\overline{(1+i)^n}+\overline{(1-i)^n}=\\ \\=(\overline{1+i})^n+(\overline{1-i})^n=\\ \\ =(1-i)^n + (1+i)^n=z\\ \\ \Rightarrow z=\overline{z} \Rightarrow z=(1+i)^n + (1-i)^n \in \mathbb{R}$

$Cateva \ dintre \ proprietatile \ conjugatului \ unui \ numar \ complex \ pe \ care \ le-am \ folosit \ sunt:\\ \\ \boxed{\overline{a+b}=\overline{a}+\overline{b}}\\ \\ \boxed{\overline{a^n}=(\overline{a})^n}$