Question 36
Let $n$ be a positive integer. Consider sums
$$
S_n = \sum_{m=0}^{n} (-1)^{j_m} m^2,
$$
where $j_m \in \{0, 1\}$. Show that it is always possible to choose the numbers $j_m$ in such a way that $0 \le S_n \le 4$.
$$
S_n = \sum_{m=0}^{n} (-1)^{j_m} m^2,
$$
where $j_m \in \{0, 1\}$. Show that it is always possible to choose the numbers $j_m$ in such a way that $0 \le S_n \le 4$.