4B. Let $z$ be a complex number such that $z+\frac{1}{z}=1$. Calculate the value of the expression

$$
z^{1993}+z^{1994}+z^{1995}+z^{1996}+z^{1997}
$$
Solution. From $z+\frac{1}{z}=1$, we obtain

$$
z^{2}+\frac{1}{z^{2}}=\left(z+\frac{1}{z}\right)^{2}-2=-1
$$

From $z+\frac{1}{z}=1$ we get $z^{2}-z+1=0$, from which it follows

$$
\begin{gathered}
\left(z^{2}-z+1\right)(z+1)=0 \\
z^{3}+1=0
\end{gathered}
$$

i.e.

$$
z^{3}=-1
$$

According to this,

$$
\begin{aligned}
z^{1993}+z^{1994}+z^{1995}+z^{1996}+z^{1997} & =z^{1995}\left(\frac{1}{z^{2}}+\frac{1}{z}+1+z+z^{2}\right) \\
& =z^{1995}=\left(z^{3}\right)^{665}=(-1)^{665}=-1
\end{aligned}
$$

## III year