A) \[\frac{1}{3}\]
B) \[\frac{5}{16}\]
C) \[\frac{3}{8}\]
D) \[\frac{41}{7280}\]
Correct Answer: B
Solution :
| Given expression |
| \[=\frac{1}{1\times 4}+\frac{1}{4\times 7}+\frac{1}{7\times 10}+\frac{1}{10\times 13}+\frac{1}{13\times 16}\] |
| \[=\frac{1}{3}\left( \begin{align} & \left( \frac{3}{1\times 4} \right)+\left( \frac{3}{4\times 7} \right)+\left( \frac{3}{7\times 10} \right)+\left( \frac{3}{10\times 13} \right)+ \\ & \left( \frac{3}{13\times 16} \right) \\ \end{align} \right)\] |
| \[=\frac{1}{3}\left[ \begin{align} & \left( 1-\frac{1}{4} \right)+\left( \frac{1}{4}-\frac{1}{7} \right)+\left( \frac{1}{7}-\frac{1}{10} \right)+\left( \frac{1}{10}-\frac{1}{13} \right) \\ & +\left( \frac{1}{13}-\frac{1}{16} \right) \\ \end{align} \right]\] |
| \[=\frac{1}{3}\left( 1-\frac{1}{4}+\frac{1}{4}-\frac{1}{7}+\frac{1}{7}-\frac{1}{10}+\frac{1}{10}-\frac{1}{13}+\frac{1}{13}-\frac{1}{16} \right)\]\[=\frac{1}{3}\left( 1-\frac{1}{16} \right)=\frac{1}{3}\left( \frac{16-1}{16} \right)=\frac{1}{3}\times \frac{15}{16}=\frac{5}{16}\] |
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