Question 103
Suppose that $p$ is an odd prime, $p \ge 7$ and $q = \frac{3p-7}{2}$.
Define the series
$$
S_q = \frac{1}{2 \cdot 3 \cdot 4} + \frac{1}{5 \cdot 6 \cdot 7} + \cdots + \frac{1}{(q+1)(q+2)(q+3)}
$$
Express $1 + 2S_q - \frac{1}{p}$ as a rational number $\frac{m}{n}$ with $(m, n) = 1$.
Prove that $m$ is a multiple of $p$.
Define the series
$$
S_q = \frac{1}{2 \cdot 3 \cdot 4} + \frac{1}{5 \cdot 6 \cdot 7} + \cdots + \frac{1}{(q+1)(q+2)(q+3)}
$$
Express $1 + 2S_q - \frac{1}{p}$ as a rational number $\frac{m}{n}$ with $(m, n) = 1$.
Prove that $m$ is a multiple of $p$.