The first and second terms of a sequence are 4 and 5 , respectively. Each term after the second is determined by increasing the previous term by one and dividing the result by the term before that. For example, the third term equals $\frac{5+1}{4}$. What is the 1234 th term of this sequence?
We write out the first few terms of the sequence using the given rule:

$$
4, \quad 5, \quad \frac{5+1}{4}=\frac{3}{2}, \quad \frac{3 / 2+1}{5}=\frac{1}{2}, \quad \frac{1 / 2+1}{3 / 2}=1, \quad \frac{1+1}{1 / 2}=4, \quad \frac{4+1}{1}=5, \quad \frac{5+1}{4}=\frac{3}{2}
$$

Since the 6 th and 7 th terms are equal to the 1 st and 2 nd terms, respectively, then the sequence repeats with period 5. (This is because two consecutive terms determine the next term, so identical pairs of consecutive terms give identical next terms.)

Since $1234=246(5)+4$, then the 1234 th term is the 4 th term after a cycle of 5 and so equals the 4 th term, which is $\frac{1}{2}$.

ANSWER: $\frac{1}{2}$