10-2-1. Petya writes down a sequence of numbers: if the current number is equal to $x$, then the next one is $\frac{1}{1-x}$. The first number in the sequence is 2. What is the five hundredth number?
Answer: -1.

Solution variant 1. Let's list the first few terms of the obtained sequence:

$$
2,-1,1 / 2,2, \ldots
$$

We see that the sequence has entered a cycle with a period of 3. Since the number 500 when divided by 3 gives a remainder of 2, the 500th term will be the same as the 2nd. That is, the five hundredth number will be equal to -1.