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

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.