10. What is the fifth term of the series \( 2 \frac{5}{6}, 3 \frac{1}{2} \), \( 4 \frac{1}{6}, 4 \frac{5}{6}, \ldots \) ? (A) \( 5 \frac{1}{6} \) (B) \( 5 \frac{1}{2} \) (C) \( 5 \frac{5}{6} \) (D) \( 6 \frac{1}{6} \) (E) \( 5 \frac{1}{4} \)

To find the fifth term of the series \( 2 \frac{5}{6}, 3 \frac{1}{2}, 4 \frac{1}{6}, 4 \frac{5}{6}, \ldots \), we first convert the mixed numbers into improper fractions: - \( 2 \frac{5}{6} = \frac{17}{6} \) - \( 3 \frac{1}{2} = \frac{7}{2} = \frac{21}{6} \) - \( 4 \frac{1}{6} = \frac{25}{6} \) - \( 4 \frac{5}{6} = \frac{29}{6} \) Next, we determine the pattern of the series. The sequence in improper fractions is: 1. \( \frac{17}{6} \) 2. \( \frac{21}{6} \) 3. \( \frac{25}{6} \) 4. \( \frac{29}{6} \) Now, we notice that each term is increasing by \( \frac{4}{6} = \frac{2}{3} \). To find the fifth term, we continue the pattern: - Fifth term = Fourth term \( + \frac{2}{3} \) - Fifth term = \( \frac{29}{6} + \frac{4}{6} = \frac{33}{6} = 5 \frac{1}{2} \) Thus, the fifth term of the series is \( 5 \frac{1}{2} \). The answer is (B) \( 5 \frac{1}{2} \).