To solve the sequence problem, we need to analyze the given terms:

The sequence is: $3 \frac{1}{2}, \; 2 \frac{1}{4}, \; 13 \frac{1}{4}, \; 12, \; \_$

We need to identify the pattern and determine the next term.

Step 1: Convert mixed numbers into improper fractions

1. $3 \frac{1}{2} = \frac{7}{2}$
2. $2 \frac{1}{4} = \frac{9}{4}$
3. $13 \frac{1}{4} = \frac{53}{4}$

The fourth term, 12, remains as is.

Step 2: Look for relationships or patterns

Observing the denominators:

• First term: $\frac{7}{2}$, denominator is 2.
• Second term: $\frac{9}{4}$, denominator is 4.
• Third term: $\frac{53}{4}$, denominator is 4.
• Fourth term: 12 can be written as $\frac{48}{4}$, denominator is 4.

From the second term onward, all terms are expressed with a denominator of 4.

Observing the numerators:

• Numerator of the first term: 7.
• Numerator of the second term: 9.
• Numerator of the third term: 53.
• Numerator of the fourth term: 48.

Changes in the numerators:

• From 7 to 9: Increase by 2.
• From 9 to 53: Increase by 44.
• From 53 to 48: Decrease by 5.

The sequence of changes in the numerators is irregular, but it suggests a rule based on position or alternating increases and decreases.

Step 3: Determine the next numerator

If the pattern alternates between large increases and small decreases:

• Following the decrease of 5 (from 53 to 48), the next term would likely involve a significant increase.

Testing the options:

• (A) $1 \frac{1}{4} = \frac{5}{4}$: Doesn't fit the pattern.
• (B) $10 \frac{3}{4} = \frac{43}{4}$: Doesn't fit the pattern.
• (C) $23 = \frac{92}{4}$: Doesn't align with the trend.
• (D) $14 \frac{1}{2} = \frac{58}{4}$: Fits the increasing pattern, aligning with $\frac{48}{4} + 10 = \frac{58}{4}$.

Step 4: Verify with $D$

• The numerator change from 48 to 58 matches the increasing trend.
• The denominator of 4 is consistent.

Thus, the next term is $14 \frac{1}{2}$, corresponding to (D).

Would you like me to elaborate on any part of this explanation?

Related Questions:

1. How do you determine a pattern when the differences aren't constant?
2. What are the steps for converting mixed numbers to improper fractions and vice versa?
3. How can you identify alternating patterns in sequences?
4. Can arithmetic sequences include fractional terms?
5. Why is it helpful to express all terms in fractions with a common denominator?

Tip: When solving sequences, always check both numerators and denominators to identify consistent changes.