Step-by-Step Solution
Key Concept: Recognize polynomial formulas for sums of squares and use partial fractions for rational summation.
Compute $\sum_{r=0}^{9} P(r) = 1^2 + 2^2 + \ldots + 10^2 = rac{10 imes 11 imes 21}{6} = 385$ and $\sum_{r=0}^{n} rac{1}{Q(r)} = \sum_{r=0}^{n} rac{2}{(r+1)(r+2)} = 2\left(1 - rac{1}{n+2}
ight) = 2$ as $n o \infty$. Then $P(n) - Q(n) = (n+1)^2 - rac{(n+1)(n+2)}{2} = rac{n(n+1)}{2}$.
Correct Answer: 3