Question : Which of the following statements is true? I. $\frac{1}{2}+\frac{1}{6}+\frac{1}{12}+\ldots \ldots \frac{1}{110}<\frac{5}{6}$ II. $\frac{1}{3}+\frac{1}{15}+\frac{1}{35}+\ldots \ldots \frac{1}{143}>\frac{7}{13}$
Option 1: Only I
Option 2: Both I and II
Option 3: Only II
Option 4: Neither I nor II
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Correct Answer: Neither I nor II
Solution : Statement I: $\frac{1}{2}+\frac{1}{6}+\frac{1}{12}+\ldots \ldots \frac{1}{110}<\frac{5}{6}$ Expand LHS $⇒\frac{1}{2}+\frac{1}{2\times{3}}+\frac{1}{3\times{4}}+\ldots \ldots \frac{1}{10\times{11}}<\frac{5}{6}$ $⇒\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+\ldots \ldots \frac{1}{10}-\frac{1}{11}<\frac{5}{6}$ $⇒1-\frac{1}{11}<\frac{5}{6}$ $⇒\frac{10}{11}<\frac{5}{6}$ which is wrong, So, statement I is incorrect. Statement II: $\frac{1}{3}+\frac{1}{15}+\frac{1}{35}+\ldots \ldots \frac{1}{143}>\frac{7}{13}$ $⇒\frac{1}{3}+\frac{1}{3\times5}+\frac{1}{5\times7}+\ldots \ldots \frac{1}{11\times13}>\frac{7}{13}$ $⇒\frac{1}{3}+\frac{2}{2}[\frac{1}{3\times5}+\frac{1}{5\times7}+\ldots \ldots \frac{1}{11\times13}]>\frac{7}{13}$ $⇒\frac{1}{3}+\frac{1}{2}[\frac{1}{3}-\frac{1}{5}+\frac{1}{5}-\frac{1}{7}+\ldots \ldots \frac{1}{11}-\frac{1}{13}]>\frac{7}{13}$ $⇒\frac{1}{3}+\frac{1}{2}[\frac{1}{3}-\frac{1}{13}]>\frac{7}{13}$ $⇒\frac{1}{3}+\frac{5}{39}>\frac{7}{13}$ $⇒\frac{6}{13}>\frac{7}{13}$ which is wrong So, statement II is incorrect. $\therefore$ Both the statement is incorrect. Hence, the correct answer is Neither I nor II.
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