Sequences & Series
Sequence and Series
star_batch_jee_advanced_2025
Grade 11
Question:
The sum $\sum_{k=1}^{n} \frac{1}{(k+1)\sqrt{k} + k\sqrt{k+1}}$ is equal to:
\frac{\sqrt{n+1}-2}{\sqrt{n+1}}
\frac{\sqrt{n+1}-1}{\sqrt{n+1}}
\frac{\sqrt{n+1}+1}{\sqrt{n+1}}
\frac{n+1}{n\sqrt{n+1}}
Step-by-Step Solution
Key Concept: Rationalizing the denominator transforms the sum into a telescoping series of reciprocals of square roots.
The sum $\sum_{k=1}^{n} \frac{1}{\sqrt{k}(\sqrt{k+1} + \sqrt{k})}$ is rationalized by multiplying by $\frac{\sqrt{k+1} - \sqrt{k}}{\sqrt{k+1} - \sqrt{k}}$ to obtain $\sum_{k=1}^{n} \frac{\sqrt{k+1} - \sqrt{k}}{\sqrt{k}\sqrt{k+1}} = \sum_{k=1}^{n} \left(\frac{1}{\sqrt{k}} - \frac{1}{\sqrt{k+1}}\right)$. This telescopes to $1 - \frac{1}{\sqrt{n+1}}$.
Correct Answer: 2