12. Simplify the expression:

\[
-(-(-(\ldots-(-1) \ldots)))
\]

which contains 200 pairs of parentheses.

If solving the problem is difficult, consider the numbers:

\[
\begin{aligned}
& -(-1)=\ldots, \\
& -(-(-1))=\ldots
\end{aligned}
\]

notice the pattern and draw a conclusion.

It is appropriate to inform students that this approach to solving problems was often used by many mathematicians, and in particular one of the greatest mathematicians in the world - Leonhard Euler (1707-1783). He writes: “... properties of numbers known today were for the most part discovered by observation and were discovered long before their truth was confirmed by rigorous proofs. There are even many properties of numbers with which we are well acquainted, but which we are still unable to prove; only observation has led us to their knowledge.”
12. -1. 13. 1) \(x=4n-1\); 2) \(y=(-1)^{n+1}\); 3) \(z=n^2+1\); 4) \(t=2^n-1\).