Sequences & Series
Summation — Ratio of Series
nta_pyq_2024_apr
Grade 11
Question:
The value of $\dfrac{1\times2^2+2\times3^2+\cdots+100\times101^2}{1^2\times2+2^2\times3+\cdots+100^2\times101}$ is
$\dfrac{32}{31}$
$\dfrac{31}{30}$
$\dfrac{306}{305}$
$\dfrac{305}{301}$
Step-by-Step Solution
Key Concept: Numerator $=\sum_{r=1}^{100}r(r+1)^2$, denominator $=\sum_{r=1}^{100}r^2(r+1)$. Factor: ratio $=\frac{\sum r(r+1)^2}{\sum r^2(r+1)}$. Simplify using standard sum formulas with $n=100$.
$\frac{\sum r(r+1)^2}{\sum r^2(r+1)}=\frac{305}{301}$.
Correct Answer: 4