6 th term of the sequence \((7 / 3),(35 / 6),[(121) /(12)]\), \([(335) /(24)], \ldots .\) is (A) \([(2113) /(96)]\) (B) \([(2112) /(96)]\) (C) \([(865) /(48)]\) (D) \([(2111) /(96)]\)

First, let's write down the given terms of the sequence and observe the relationship between numerators and denominators of each term. Term 1: \(\frac{7}{3}\) Term 2: \(\frac{35}{6}\) Term 3: \(\frac{121}{12}\) Term 4: \(\frac{335}{24}\) On observing the numerators \(7, 35, 121, 335\), we can observe a pattern that each numerator is obtained by multiplying the previous numerator by its immediate index. \(\text{Term 2's numerator} = 7 \cdot (2) = 35\) \(\text{Term 3's numerator} = 35 \cdot (3) = 121\) \(\text{Term 4's numerator} = 121 \cdot (4) = 335\) The denominators of the terms are obtained by multiplying 3 by the index of each term. \(\text{Term 1's denominator} = 3 \cdot (1) = 3\) \(\text{Term 2's denominator} = 3 \cdot (2) = 6\) \(\text{Term 3's denominator} = 3 \cdot (4) = 12\) \(\text{Term 4's denominator} = 3 \cdot (8) = 24\) Notice that the multiples are incremented by \(2\): \(2, 4, 8,\ldots\).

To find the \(6^{th}\) term's numerator, we can simply multiply the \(5^{th}\) term's numerator (which is the \(4^{th}\) term's numerator multiplied by \(5\)) by \(6\): \(5^{th} \text{ term's numerator} = 335 \cdot (5) = 1675\) \(6^{th} \text{ term's numerator} = 1675 \cdot (6) = 10050\) For the \(6^{th}\) term's denominator, we can multiply the factor \(3\) by the next number in the sequence of multiples, i.e. the next element in sequence \(2, 4, 8,\ldots\). The sequence is formed by doubling the previous term, so the next term is \(8 \cdot 2 = 16\). \(6^{th} \text{ term's denominator} = 3 \cdot (16) = 48\)

Now we can write out the \(6^{th}\) term's fraction and simplify it: \[\frac{10050}{48}\] To simplify, we can divide both the numerator and the denominator by the greatest common divisor (GCD) of \(10050\) and \(48\), which is \(6\). \[\frac{10050 \div 6}{48 \div 6} = \frac{1675}{8}\] Thus, the \(6^{th}\) term of the sequence is: \[\frac{1675}{8}\] Since this number doesn't match any of the given options (A), (B), (C), or (D), there must be a mistake in the given options.