Question 97
Suppose that the sequence of positive numbers $x_1, x_2, \ldots, x_n, \ldots$ satisfies $(8x_2 - 7x_1)x_1^7 = 8$ and
$$
x_{k+1}x_{k-1} - x_k^2 = \frac{x_{k-1}^8 - x_k^8}{(x_kx_{k-1})^7}, \quad k \ge 2.
$$
Find positive real number $a$ such that when $x_1 \gt a$ one has $x_1 \gt x_2 \gt \cdots \gt x_n \gt \cdots$, and when $0 \lt x_1 \lt a$ one does not have such monotonicity. (Posed by Li Shenghong)
$$
x_{k+1}x_{k-1} - x_k^2 = \frac{x_{k-1}^8 - x_k^8}{(x_kx_{k-1})^7}, \quad k \ge 2.
$$
Find positive real number $a$ such that when $x_1 \gt a$ one has $x_1 \gt x_2 \gt \cdots \gt x_n \gt \cdots$, and when $0 \lt x_1 \lt a$ one does not have such monotonicity. (Posed by Li Shenghong)