The sequence of numbers $t_{1}, t_{2}, t_{3}, \ldots$ is defined by

$$
\left\{\begin{array}{l}
t_{1}=2 \\
t_{n+1}=\frac{t_{n}-1}{t_{n}+1}
\end{array}\right.
$$

Suggestion: Calculate the first five terms of the sequence.

for each positive integer $n$. Find $\mathrm{t}_{2011}$.
Let's calculate the first terms of the sequence:

$$
\begin{aligned}
& \mathrm{t}_{2}=\frac{2-1}{2+1}=\frac{1}{3} \\
& \mathrm{t}_{3}=\frac{1 / 3-1}{1 / 3+1}=-\frac{1}{2} \\
& \mathrm{t}_{4}=\frac{-1 / 2-1}{-1 / 2+1}=-3 \\
& \mathrm{t}_{5}=\frac{-3-1}{-3+1}=2
\end{aligned}
$$

Thus, the first five terms of the sequence are $2, 1/3, -1/2, -3$, and 2. We observe that the sequence repeats every 4 terms, that is,

$$
2=\mathrm{t}_{1}=\mathrm{t}_{5}=\mathrm{t}_{9}=\mathrm{t}_{13}=\cdots=\mathrm{t}_{2009}
$$

Thus, $\mathrm{t}_{2010}=1/3$ and $\mathrm{t}_{2011}=-1/2$.

## Related Problem

Dodo's calculator has a special key with the symbol $\mathrm{R}$. If the display shows a number $x$ different from 2, pressing $\left\ulcorner\right.$ displays the value of $\frac{2 x-3}{x-2}$.

(a) If Dodo puts 4 on the display and presses $乃$, what number will appear?

(b) Dodo put a number on the display and, upon pressing $乃$, the same number appeared. What numbers could he have put on the display?

(c) Dodo noticed that, by putting 4 on the display and pressing $r$ twice, 4 appears again; similarly, by putting 5 and pressing $\beta$ twice, 5 appears again. Will the same happen for any number different from 2? Explain.

Hint: The common ratio of the geometric progression must be less than 2.