Question 5
Let $q = \frac{3p-5}{2}$ where $p$ is an odd prime, and let
$$ S_q = \frac{1}{2\cdot 3 \cdot 4} + \frac{1}{5\cdot 6 \cdot 7}
+ \ldots + \frac{1}{q(q+1)(q+2)}. $$
Prove that if $\frac{1}{p}-2S_q = \frac{m}{n}$ for integers $m$ and $n$,
then $m - n$ is divisible by $p$.
Proposed by Titu Andreescu (credit in the supplied third-party notes).
$$ S_q = \frac{1}{2\cdot 3 \cdot 4} + \frac{1}{5\cdot 6 \cdot 7}
+ \ldots + \frac{1}{q(q+1)(q+2)}. $$
Prove that if $\frac{1}{p}-2S_q = \frac{m}{n}$ for integers $m$ and $n$,
then $m - n$ is divisible by $p$.
Proposed by Titu Andreescu (credit in the supplied third-party notes).