Question 85
Let $p$ be an odd prime and $S_{q} = \frac{1}{2 \cdot 3 \cdot 4} + \frac{1}{5 \cdot 6 \cdot 7} + \ldots + \frac{1}{q(q+1)(q+2)}$, where $q = \frac{3p-5}{2}$. We write $\frac{1}{p} - 2 S_{q}$ in the form $\frac{m}{n}$, where $m$ and $n$ are integers. Prove that $m \equiv n \pmod{p}$; that is, that $m$ and $n$ leave the same remainder when divided by $p$.