5. Considering the arithmetic sequence: \( 11 \frac{1}{2}, 9 \frac{5}{16}, 7 \frac{1}{8}, \ldots \) If the first three terms are shown, what is the fifth term?

To find the fifth term of the arithmetic sequence given by the first three terms \( 11 \frac{1}{2}, 9 \frac{5}{16}, 7 \frac{1}{8} \), we first need to determine the common difference of the sequence. 1. **Convert the mixed numbers to improper fractions:** - The first term \( 11 \frac{1}{2} \) can be converted as follows: \[ 11 \frac{1}{2} = \frac{11 \times 2 + 1}{2} = \frac{22 + 1}{2} = \frac{23}{2} \] - The second term \( 9 \frac{5}{16} \) can be converted as follows: \[ 9 \frac{5}{16} = \frac{9 \times 16 + 5}{16} = \frac{144 + 5}{16} = \frac{149}{16} \] - The third term \( 7 \frac{1}{8} \) can be converted as follows: \[ 7 \frac{1}{8} = \frac{7 \times 8 + 1}{8} = \frac{56 + 1}{8} = \frac{57}{8} \] 2. **Calculate the common difference \( d \):** - The common difference can be found by subtracting the first term from the second term: \[ d = 9 \frac{5}{16} - 11 \frac{1}{2} = \frac{149}{16} - \frac{23}{2} \] - To perform this subtraction, we need a common denominator. The least common multiple of 16 and 2 is 16. Thus, we convert \( \frac{23}{2} \) to sixteenths: \[ \frac{23}{2} = \frac{23 \times 8}{2 \times 8} = \frac{184}{16} \] - Now we can subtract: \[ d = \frac{149}{16} - \frac{184}{16} = \frac{149 - 184}{16} = \frac{-35}{16} \] 3. **Find the fifth term:** - The fifth term can be calculated using the formula for the \( n \)-th term of an arithmetic sequence: \[ a_n = a_1 + (n-1)d \] - Here, \( a_1 = \frac{23}{2} \), \( d = \frac{-35}{16} \), and \( n = 5 \): \[ a_5 = \frac{23}{2} + (5-1) \cdot \frac{-35}{16} \] - Simplifying this: \[ a_5 = \frac{23}{2} + 4 \cdot \frac{-35}{16} = \frac{23}{2} - \frac{140}{16} \] - Convert \( \frac{23}{2} \) to sixteenths: \[ \frac{23}{2} = \frac{23 \times 8}{2 \times 8} = \frac{184}{16} \] - Now we can subtract: \[ a_5 = \frac{184}{16} - \frac{140}{16} = \frac{184 - 140}{16} = \frac{44}{16} = \frac{11}{4} \] 4. **Convert back to a mixed number:** - The fifth term \( \frac{11}{4} \) can be expressed as a mixed number: \[ \frac{11}{4} = 2 \frac{3}{4} \] Thus, the fifth term of the arithmetic sequence is \( 2 \frac{3}{4} \).