1. Let

$$
f(n)=\left(\frac{1+i}{\sqrt{2}}\right)^{n}+\left(\frac{1-i}{\sqrt{2}}\right)^{n}
$$

where $i$ is the imaginary unit. Calculate $f(1986)+f(1990)$!
Solution. If $z_{1}=\frac{1+i}{\sqrt{2}}, z_{2}=\frac{1-i}{\sqrt{2}}$, then $z_{1}^{2}=i, z_{2}^{2}=-i, z_{1}^{4}=z_{2}^{4}=-1$. Using this, we get:

$$
\begin{aligned}
f(1986)+f(1990) & =z_{1}^{1986}+z_{2}^{1986}+z_{1}^{1990}+z_{2}^{1990} \\
& =z_{1}^{1986}\left(1+z_{1}^{4}\right)+z_{2}^{1986}\left(1+z_{2}^{4}\right) \\
& =z_{1}^{1986}(1-1)+z_{2}^{1986}(1-1)=0
\end{aligned}
$$