Question 12
$a_1, a_2, \ldots, a_n$ are real numbers. Let $b_k = (a_1 + a_2 + \ldots + a_k)/k$ for $k = 1, 2, \ldots, n$. Let
$$
C = (a_1 - b_1)^2 + (a_2 - b_2)^2 + \ldots + (a_n - b_n)^2,
$$
and
$$
D = (a_1 - b_n)^2 + (a_2 - b_n)^2 + \ldots + (a_n - b_n)^2.
$$
Show that $C \leq D \leq 2C$.
$$
C = (a_1 - b_1)^2 + (a_2 - b_2)^2 + \ldots + (a_n - b_n)^2,
$$
and
$$
D = (a_1 - b_n)^2 + (a_2 - b_n)^2 + \ldots + (a_n - b_n)^2.
$$
Show that $C \leq D \leq 2C$.