A) \[\frac{5}{39}\]
B) \[\frac{4}{39}\]
C) \[\frac{2}{39}\]
D) \[\frac{7}{39}\]
Correct Answer: A
Solution :
\[\frac{1}{15}+\frac{1}{35}+\frac{1}{63}+\frac{1}{99}+\frac{1}{143}\] \[=\frac{1}{3\times 5}\times \frac{1}{5\times 7}+\frac{1}{7\times 9}\] \[+\frac{1}{9\times 11}+\frac{1}{11\times 13}\] \[=\frac{1}{2}\left( \frac{1}{3}-\frac{1}{5}+\frac{1}{5}-\frac{1}{7}+\frac{1}{7}-\frac{1}{9}+\frac{1}{9}-\frac{1}{11}+\frac{1}{11}-\frac{1}{13} \right)\] \[=\frac{1}{2}\left( \frac{1}{3}-\frac{1}{13} \right)=\frac{1}{2}\left( \frac{13-3}{39} \right)=\frac{5}{39}\]
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