Question 154
Let $a_0, a_1, a_2, \dots$ be a sequence of real numbers such that $a_0 = 0$, $a_1 = 1$, and for every $n \ge 2$ there exists $1 \le k \le n$ satisfying
$$
a_n = \frac{a_{n-1} + \dots + a_{n-k}}{k}.
$$
Find the maximal possible $a_{2018} - a_{2017}$.
$$
a_n = \frac{a_{n-1} + \dots + a_{n-k}}{k}.
$$
Find the maximal possible $a_{2018} - a_{2017}$.