Question 165
Let $\{a_n\}$ and $\{b_n\}$ be two sequences of positive real numbers such that, for any positive integer $n$,
$$
a_{n+1} = a_n - \frac{1}{1 + \sum_{i=1}^{n} \frac{1}{a_i}}, \quad \text{and} \quad b_{n+1} = b_n + \frac{1}{1 + \sum_{i=1}^{n} \frac{1}{b_i}}.
$$
(1)
If $a_{100}b_{100} = a_{101}b_{101}$, find the value of $a_1 - b_1$.
(2)
If $a_{100} = b_{99}$, which one of $a_{100} + b_{100}$ and $a_{101} + b_{101}$ is larger?
$$
a_{n+1} = a_n - \frac{1}{1 + \sum_{i=1}^{n} \frac{1}{a_i}}, \quad \text{and} \quad b_{n+1} = b_n + \frac{1}{1 + \sum_{i=1}^{n} \frac{1}{b_i}}.
$$
(1)
If $a_{100}b_{100} = a_{101}b_{101}$, find the value of $a_1 - b_1$.
(2)
If $a_{100} = b_{99}$, which one of $a_{100} + b_{100}$ and $a_{101} + b_{101}$ is larger?