1. Simplify the expression

$$
1-a+a^{2}-a^{3}+\ldots+a^{1980}-\frac{a^{1981}}{1+a}
$$

and find its value for $a=-\frac{4}{5}$.
Solution. Using the formula

$$
x^{n}+y^{n}=(x+y)\left(x^{n-1}-y x^{n-2}+\ldots+(-1)^{n-1} y^{n-1}\right)
$$

we get

$$
\begin{aligned}
1-a+a^{2}-a^{3}+\ldots+a^{1980}-\frac{a^{1981}}{1+a} & =\frac{\left(1-a+a^{2}-a^{3}+\ldots+a^{1980}\right)(1+a)-a^{1981}}{1+a} \\
& =\frac{1+a^{1981}-a^{1981}}{1+a}=\frac{1}{1+a}
\end{aligned}
$$

for $a=-\frac{4}{5}$, the value of the expression is 5.