Question 18
Let $\{a_n\}_{n=1}^{\infty}$ be a sequence of positive integers such that for every positive integer $n$
$$
a_{n+1} = (n + 1)(a_n - n + 1).
$$
In terms of $a_1$, determine the greatest positive integer $k$ such that $\gcd(a_i, a_{i+1}) = k$
for some positive integer $i \ge 2$. (Note that $\gcd(x, y)$ denotes the greatest common
divisor of integers $x$ and $y$.)
$$
a_{n+1} = (n + 1)(a_n - n + 1).
$$
In terms of $a_1$, determine the greatest positive integer $k$ such that $\gcd(a_i, a_{i+1}) = k$
for some positive integer $i \ge 2$. (Note that $\gcd(x, y)$ denotes the greatest common
divisor of integers $x$ and $y$.)