1. Zadatak

Primijetimo da je

$$\begin{aligned} z&=\frac{1+i}{\sqrt2},&z^2&=i,&z^3&=\frac{-1+i}{\sqrt2},&z^4&=-1,\\ z^5&=\frac{-1-i}{\sqrt2},&z^6&=-i,&z^7&=\frac{1-i}{\sqrt2},&z^8&=1, \end{aligned}$$

pa je

$$1+z+z^2+\ldots+z^7=0.$$

Kako je $z^k=z^{8l+k}$, $k\in\{0,1,2,\ldots,7\}$, $l\in\mathbb N$, i kako je $2006=8\cdot250+6$, dani zbroj $S$ možemo napisati u obliku

$$\begin{aligned} S&=(1+z+z^2+\ldots+z^7)(1+z^8+z^{2\cdot8}+\ldots+z^{249\cdot8})\\ &\quad +(1+z+z^2+\ldots+z^6)\cdot z^{250\cdot8}\\ &=1+z+z^2+\ldots+z^6=\frac{-1+i}{\sqrt2}. \end{aligned}$$