Question 77
Find all functions $f: \mathbb{N} \rightarrow (0, \infty)$ such that $f(4) = 4$ and
$$
\frac{1}{f(1) f(2)} + \frac{1}{f(2) f(3)} + \cdots + \frac{1}{f(n) f(n+1)} = \frac{f(n)}{f(n+1)}, \quad \forall n \in \mathbb{N},
$$
where $\mathbb{N} = \{1, 2, \ldots\}$ is the set of positive integers.
$$
\frac{1}{f(1) f(2)} + \frac{1}{f(2) f(3)} + \cdots + \frac{1}{f(n) f(n+1)} = \frac{f(n)}{f(n+1)}, \quad \forall n \in \mathbb{N},
$$
where $\mathbb{N} = \{1, 2, \ldots\}$ is the set of positive integers.