Question 171
Find all real numbers $x$ that can be written as
$$
x = \frac{a_{0}}{a_{1} a_{2} \ldots a_{n}} + \frac{a_{1}}{a_{2} a_{3} \ldots a_{n}} + \frac{a_{2}}{a_{3} a_{4} \ldots a_{n}} + \ldots + \frac{a_{n-2}}{a_{n-1} a_{n}} + \frac{a_{n-1}}{a_{n}},
$$
where $n, a_{1}, a_{2}, \ldots, a_{n}$ are positive integers and
$$
1 = a_{0} \leq a_{1} \lt a_{2} \lt \ldots \lt a_{n}.
$$
$$
x = \frac{a_{0}}{a_{1} a_{2} \ldots a_{n}} + \frac{a_{1}}{a_{2} a_{3} \ldots a_{n}} + \frac{a_{2}}{a_{3} a_{4} \ldots a_{n}} + \ldots + \frac{a_{n-2}}{a_{n-1} a_{n}} + \frac{a_{n-1}}{a_{n}},
$$
where $n, a_{1}, a_{2}, \ldots, a_{n}$ are positive integers and
$$
1 = a_{0} \leq a_{1} \lt a_{2} \lt \ldots \lt a_{n}.
$$