Exercise 4. Let $a_{1}, a_{2}, a_{3}, \ldots$ be a sequence of numbers such that $a_{1}=2$, $a_{2}=3$, and $a_{n}=\frac{a_{n-1}}{a_{n-2}}$ for all integers $n \geqslant 3$. For example, $a_{3}=\frac{a_{2}}{a_{1}}=\frac{3}{2}$.

Determine the value of $a_{2014}$.
Solution to Exercise 4 We have

$$
a_{4}=\frac{a_{3}}{a_{2}}=\frac{1}{2}, \quad a_{5}=\frac{a_{4}}{a_{3}}=\frac{1}{3}, \quad a_{6}=\frac{2}{3}, \quad a_{7}=2=a_{1}, \quad a_{8}=3=a_{2}, \quad a_{9}=\frac{3}{2}=a_{3}
$$

and so on. We see that the same six terms repeat. Since $2014=4+6 \times 335$, we have $a_{2014}=a_{4}=\frac{1}{2}$.