Question 33
A sequence $a_0, a_1, a_2, \dots$ of positive real numbers is given such that
$$
a_1 = 1 - a_0, \quad a_{n+1} = 1 - a_n(1 - a_n) \quad \text{for all } n \ge 1.
$$
Prove that
$$
a_0 a_1 \cdots a_n \left( \frac{1}{a_0} + \frac{1}{a_1} + \cdots + \frac{1}{a_n} \right) = 1
$$
holds for every positive integer $n$.
$$
a_1 = 1 - a_0, \quad a_{n+1} = 1 - a_n(1 - a_n) \quad \text{for all } n \ge 1.
$$
Prove that
$$
a_0 a_1 \cdots a_n \left( \frac{1}{a_0} + \frac{1}{a_1} + \cdots + \frac{1}{a_n} \right) = 1
$$
holds for every positive integer $n$.