Question : Evaluate: $\frac{1}{15}+\frac{1}{35}+\frac{1}{63}+\frac{1}{99}+\frac{1}{143}$
Option 1: $\frac{4}{39}$
Option 2: $\frac{5}{39}$
Option 3: $\frac{10}{39}$
Option 4: $\frac{7}{39}$
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Correct Answer: $\frac{5}{39}$
Solution :
$\frac{1}{15}+\frac{1}{35}+\frac{1}{63}+\frac{1}{99}+\frac{1}{143}$
$=\frac{1}{(3×5)}+\frac{1}{(5×7)}+\frac{1}{(7×9)}+\frac{1}{(9×11)}+\frac{1}{(11×13)}$
$= \frac{1}{2}×[( \frac{1}{3}− \frac{1}{5})+( \frac{1}{5}− \frac{1}{7})+( \frac{1}{7}− \frac{1}{9})+( \frac{1}{9}− \frac{1}{11})+( \frac{1}{11}− \frac{1}{13})]$
$= \frac{1}{2}×( \frac{1}{3}− \frac{1}{13})$
$= \frac{1}{2}×( \frac{13-3}{39})$
$= \frac{5}{39}$
Hence, the correct answer is $ \frac{5}{39}$.
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