Question 81
Determine the largest real number $M$ such that for each infinite sequence $x_0, x_1, x_2, \dots$ of real numbers satisfying
a. $x_0 = 1$ and $x_1 = 3$,
b. $x_0 + x_1 + \dots + x_{n-1} \ge 3x_n - x_{n+1}$ for all $n \ge 1$,
the inequality
$$
\frac{x_{n+1}}{x_n} \gt M,
$$
holds for all $n \ge 0$.
a. $x_0 = 1$ and $x_1 = 3$,
b. $x_0 + x_1 + \dots + x_{n-1} \ge 3x_n - x_{n+1}$ for all $n \ge 1$,
the inequality
$$
\frac{x_{n+1}}{x_n} \gt M,
$$
holds for all $n \ge 0$.