Question 72
Let $d$ be a positive integer. Show that for every integer $S$, there exists an integer $n \gt 0$ and a sequence $\epsilon_1, \epsilon_2, \dots, \epsilon_n$, where for any $k$, $\epsilon_k = 1$ or $\epsilon_k = -1$, such that
$$
S = \epsilon_1(1+d)^2 + \epsilon_2(1+2d)^2 + \epsilon_3(1+3d)^2 + \dots + \epsilon_n(1+nd)^2.
$$
$$
S = \epsilon_1(1+d)^2 + \epsilon_2(1+2d)^2 + \epsilon_3(1+3d)^2 + \dots + \epsilon_n(1+nd)^2.
$$