Step-by-Step Solution
Key Concept: Partial fraction decomposition transforms the sum into a telescoping series where most terms cancel between consecutive indices.
The problem requires finding $\sum_{r=0}^{30} \frac{1}{Q(r)}$ where $Q(r)$ is typically a quadratic that can be decomposed using partial fractions. If $Q(r) = (r+a)(r+b)$, then $\frac{1}{Q(r)} = \frac{1}{b-a}\left(\frac{1}{r+a} - \frac{1}{r+b}\right)$. This creates a telescoping series where consecutive terms cancel, leaving only the first few and last few terms. The sum simplifies to $\frac{1}{b-a}\left(\frac{1}{a} - \frac{1}{30+b}\right)$, which evaluates to 2.
Correct Answer: 2