Question #40702

Let $S_k(n) = \sum_{i=1}^n i^k$ for $k \in \{1, 2, 3\}$. It is known that $S_1(n) = a_1 n^2 + b_1 n + c_1$, $S_2(n) = a_2 n^3 + b_2 n^2 + c_2 n$, and $S_3(n) = a_3 n^4 + b_3 n^3 + c_3 n^2$. Let $M$ be the $3 \times 3$ matrix defined by $M = \begin{bmatrix} a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3 \end{bmatrix}$. Consider the system of linear equations $M \begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} \alpha \\ \beta \\ \gamma \end{bmatrix}$. If this system has infinitely many solutions, which of the following relations must hold true?