Two positive numbers, a and b, are in the sequence 4. a,b,12. The first three numbers form a geometric sequence, and the last three numbers form an arithmetic sequence. The difference b-a equals

Options

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

Explanation

To solve the problem, we need to find the difference \( b - a \) given that the numbers \( 4, a, b, 12 \) form a geometric sequence for the first three numbers and an arithmetic sequence for the last three numbers.

### Step-by-Step Solution:

1. **Identify the properties of the sequences:**
- The first three numbers \( 4, a, b \) form a geometric sequence. This means that the ratio of consecutive terms is constant.
- The last three numbers \( a, b, 12 \) form an arithmetic sequence. This means that the difference between consecutive terms is constant.

2. **Set up the equations for the geometric sequence:**
- For the geometric sequence \( 4, a, b \), we have:
\[ \frac{a}{4} = \frac{b}{a} \]
- Cross-multiplying gives:
\[ a^2 = 4b \quad \text{(Equation 1)} \]

3. **Set up the equations for the arithmetic sequence:**
- For the arithmetic sequence \( a, b, 12 \), we have:
\[ b - a = 12 - b \]
- Rearranging gives:
\[ 2b = a + 12 \quad \text{(Equation 2)} \]

4. **Substitute Equation 1 into Equation 2:**
- From Equation 1, we know \( b = \frac{a^2}{4} \).
- Substitute \( b \) into Equation 2:
\[ 2\left(\frac{a^2}{4}\right) = a + 12 \]
- Simplifying gives:
\[ \frac{a^2}{2} = a + 12 \]
- Multiplying through by 2 to eliminate the fraction:
\[ a^2 = 2a + 24 \]
- Rearranging gives:
\[ a^2 - 2a - 24 = 0 \quad \text{(Equation 3)} \]

5. **Solve the quadratic equation (Equation 3):**
- We can factor the quadratic:
\[ (a - 6)(a + 4) = 0 \]
- This gives us two potential solutions:
\[ a = 6 \quad \text{or} \quad a = -4 \]
- Since \( a \) must be positive, we take \( a = 6 \).

6. **Find \( b \) using the value of \( a \):**
- Substitute \( a = 6 \) back into Equation 1:
\[ b = \frac{6^2}{4} = \frac{36}{4} = 9 \]

7. **Calculate \( b - a \):**
- Now we can find the difference:
\[ b - a = 9 - 6 = 3 \]

### Final Answer:
The difference \( b - a \) is \( 3 \).