If a, b & 3c are in arithmetic progression and a, b & 4c are in geometric progression, then the possible value of `(a)/(b)` are

Options

A. `{(2)/(3),2}`
B. `{(3)/(2),(1)/(2)}`
C. `{(2)/(3),(3)/(2)}`
D. `{(1)/(2),2}`

Answer

To solve the problem, we need to find the possible values of \(\frac{a}{b}\) given that \(a\), \(b\), and \(3c\) are in arithmetic progression (AP) and \(a\), \(b\), and \(4c\) are in geometric progression (GP).

### Step-by-Step Solution:

1. **Understanding the Conditions**:
- For \(a\), \(b\), and \(3c\) to be in AP, we have:
\[ 2b = a + 3c \quad \text{(Equation 1)} \]
- For \(a\), \(b\), and \(4c\) to be in GP, we have:
\[ b^2 = a \cdot 4c \quad \text{(Equation 2)} \]

2. **Express \(c\) from Equation 1**:
Rearranging Equation 1 gives:
\[ 3c = 2b - a \implies c = \frac{2b - a}{3} \]

3. **Substituting \(c\) into Equation 2**:
Substitute \(c\) from the previous step into Equation 2:
\[ b^2 = a \cdot 4 \left(\frac{2b - a}{3}\right) \]
Simplifying this gives:
\[ b^2 = \frac{4a(2b - a)}{3} \]
Multiplying both sides by 3 to eliminate the fraction:
\[ 3b^2 = 4a(2b - a) \]

4. **Expanding and Rearranging**:
Expanding the right-hand side:
\[ 3b^2 = 8ab - 4a^2 \]
Rearranging gives us a quadratic equation in terms of \(b\):
\[ 3b^2 - 8ab + 4a^2 = 0 \]

5. **Using the Quadratic Formula**:
The quadratic formula is given by:
\[ b = \frac{-B \pm \sqrt{B^2 - 4AC}}{2A} \]
Here, \(A = 3\), \(B = -8a\), and \(C = 4a^2\). Plugging these into the formula:
\[ b = \frac{8a \pm \sqrt{(-8a)^2 - 4 \cdot 3 \cdot 4a^2}}{2 \cdot 3} \]
Simplifying the discriminant:
\[ b = \frac{8a \pm \sqrt{64a^2 - 48a^2}}{6} = \frac{8a \pm \sqrt{16a^2}}{6} = \frac{8a \pm 4a}{6} \]

6. **Finding the Values of \(b\)**:
This gives us two possible values for \(b\):
- \(b = \frac{12a}{6} = 2a\)
- \(b = \frac{4a}{6} = \frac{2a}{3}\)

7. **Finding \(\frac{a}{b}\)**:
Now, we can find \(\frac{a}{b}\) for both cases:
- For \(b = 2a\):
\[ \frac{a}{b} = \frac{a}{2a} = \frac{1}{2} \]
- For \(b = \frac{2a}{3}\):
\[ \frac{a}{b} = \frac{a}{\frac{2a}{3}} = \frac{3}{2} \]

### Final Values:
The possible values of \(\frac{a}{b}\) are:
\[ \frac{1}{2} \quad \text{and} \quad \frac{3}{2} \]