A) \[\frac{1}{4}\]
B) \[\frac{1}{2}\]
C) 1
D) \[\frac{1}{8}\]
Correct Answer: A
Solution :
[a] \[\frac{kn}{n+1}=\left[ \frac{1}{2.4}+\frac{1}{4.6}+\frac{1}{6.8}+...n\,terms \right]\] \[=\frac{1}{2}\left[ \frac{4-2}{2.4}+\frac{6-4}{4.6}+\frac{8-6}{6.8}+...+\frac{2n+2-2n}{2n(2n+2)} \right]\] \[=\frac{1}{2}\left[ \frac{1}{2}-\frac{1}{4}+\frac{1}{4}-\frac{1}{6}+\frac{1}{6}-\frac{1}{8}+...+\frac{1}{2n}-\frac{1}{2n+2} \right]\] \[=\frac{1}{2}\left[ \frac{1}{2}-\frac{1}{2(n+1)} \right]=\frac{n}{4(n+1)}\Rightarrow k=\frac{1}{4}\]
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