The next terms of the sequence `1/2,3 1/4, 6, 8 3/4`….. Is :

Options

A. `10 1/4`
B. `10 3/4`
C. `11 1/4`
D. `11 1/2`

Answer

To find the next term in the sequence \( \frac{1}{2}, 3 \frac{1}{4}, 6, 8 \frac{3}{4} \), we will first convert all terms into improper fractions for easier calculations.

1. **Convert Mixed Numbers to Improper Fractions**:
- \( \frac{1}{2} = \frac{1}{2} \)
- \( 3 \frac{1}{4} = 3 \times 4 + 1 = \frac{12 + 1}{4} = \frac{13}{4} \)
- \( 6 = \frac{6 \times 1}{1} = \frac{6}{1} = \frac{24}{4} \)
- \( 8 \frac{3}{4} = 8 \times 4 + 3 = \frac{32 + 3}{4} = \frac{35}{4} \)

So, the sequence in improper fractions is:
\[ \frac{1}{2}, \frac{13}{4}, \frac{24}{4}, \frac{35}{4} \]

2. **Identify the Differences**:
- The difference between the second term and the first term:
\[ \frac{13}{4} - \frac{1}{2} = \frac{13}{4} - \frac{2}{4} = \frac{11}{4} \]
- The difference between the third term and the second term:
\[ \frac{24}{4} - \frac{13}{4} = \frac{24 - 13}{4} = \frac{11}{4} \]
- The difference between the fourth term and the third term:
\[ \frac{35}{4} - \frac{24}{4} = \frac{35 - 24}{4} = \frac{11}{4} \]

The differences between consecutive terms are consistent and equal to \( \frac{11}{4} \).

3. **Find the Next Term**:
- To find the next term, we add \( \frac{11}{4} \) to the last term \( \frac{35}{4} \):
\[ \text{Next term} = \frac{35}{4} + \frac{11}{4} = \frac{35 + 11}{4} = \frac{46}{4} \]

4. **Convert Back to Mixed Number**:
- Now, we simplify \( \frac{46}{4} \):
\[ \frac{46}{4} = 11 \frac{1}{2} \]

Thus, the next term in the sequence is \( 11 \frac{1}{2} \).