The next term 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 can start by converting all terms to improper fractions for easier calculations.

### Step 1: Convert mixed numbers to improper fractions
1. \( \frac{1}{2} = \frac{1}{2} \)
2. \( 3 \frac{1}{4} = 3 \times 4 + 1 = \frac{12 + 1}{4} = \frac{13}{4} \)
3. \( 6 = \frac{6 \times 1}{1} = \frac{6}{1} = \frac{24}{4} \)
4. \( 8 \frac{3}{4} = 8 \times 4 + 3 = \frac{32 + 3}{4} = \frac{35}{4} \)

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

### Step 2: Find the pattern in the numerators
Now, let's look at the numerators of these fractions:
- The numerators are: \( 1, 13, 24, 35 \)

### Step 3: Find the differences between the numerators
Next, we find the differences between consecutive numerators:
1. \( 13 - 1 = 12 \)
2. \( 24 - 13 = 11 \)
3. \( 35 - 24 = 11 \)

The differences are \( 12, 11, 11 \).

### Step 4: Identify the next difference
The pattern in the differences suggests that after the initial difference of 12, the next differences are constant at 11. Therefore, we can assume the next difference will also be 11.

### Step 5: Calculate the next numerator
To find the next numerator:
\[ 35 + 11 = 46 \]

### Step 6: Convert back to a mixed number
Now, we need to convert the next term back to a mixed number. Since we are using the denominator of 4:
\[ \frac{46}{4} = 11 \frac{2}{4} = 11 \frac{1}{2} \]

### Conclusion
Thus, the next term in the sequence is:
\[ \boxed{11 \frac{1}{2}} \]

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