Sequences & Series
Sequences And Series
nta_abhyas_2025
Grade 11
Question:
Let the sum $\sum_{n=1}^{100} \frac{1}{n(n+1)(n+2)}$ written in the rational form be $\frac{p}{q}$ (where $p$ and $q$ are co-prime), then the value of $\lfloor \frac{5q}{4p} \rfloor$ is (where $\lfloor \cdot \rfloor$ is the greatest integer function)
Step-by-Step Solution
Key Concept: Partial fractions decomposition allows a telescoping series where most terms cancel.
The sum telescopes: $\sum_{k=1}^{n} \frac{2}{n(n+1)(n+2)} = \frac{1}{2}\left(\frac{1}{1 \cdot 2} - \frac{1}{110 \cdot 111}\right) = \frac{1}{2}\left(\frac{1}{2} - \frac{1}{12210}\right) = \frac{1}{2} \cdot \frac{12208}{24420} = \frac{110}{220}$. Therefore $q - p = 110 - 27 = 83$.
Correct Answer: 83