Question 35
The sequence $x_{1}, x_{2}, x_{3}, \ldots$ is defined by $x_{1} = 2022$ and $x_{n + 1} = 7x_{n} + 5$ for all positive integers $n$. Determine the maximum positive integer $m$ such that
$$
\frac{x_{n}(x_{n}-1)(x_{n}-2)\ldots(x_{n}-m+1)}{m!}
$$
is never a multiple of $7$ for any positive integer $n$.
$$
\frac{x_{n}(x_{n}-1)(x_{n}-2)\ldots(x_{n}-m+1)}{m!}
$$
is never a multiple of $7$ for any positive integer $n$.