Question 19
Let $a_{1} + a_{2} + \ldots + a_{n} = 0$ and $|a_{1}| + |a_{2}| + \ldots + |a_{n}| = 1$.
Prove that
$$
\left|a_{1} + 2 a_{2} + \ldots + n a_{n}\right| \leq \frac{n-1}{2}
$$
Prove that
$$
\left|a_{1} + 2 a_{2} + \ldots + n a_{n}\right| \leq \frac{n-1}{2}
$$