If `x, 2x+ 2, 3x + 3 `are in `G.P. ,` then the fourth term is (A) `27` (B) `-27` (C)`13.5` (D)`-13.5`

Options

A. `27`
B. `-27`
C. `13.5`
D. `-13.5`

Explanation

To solve the problem, we need to determine if the terms \(x\), \(2x + 2\), and \(3x + 3\) are in a geometric progression (G.P.). If they are, we will find the fourth term of the G.P.

### Step-by-Step Solution:

1. **Identify the terms**: We have the terms as follows:
- First term: \(a = x\)
- Second term: \(b = 2x + 2\)
- Third term: \(c = 3x + 3\)

2. **Use the property of G.P.**: For three terms to be in G.P., the square of the middle term must be equal to the product of the other two terms. This can be expressed mathematically as:
\[ (2x + 2)^2 = x(3x + 3) \]

3. **Expand both sides**:
- Left side:
\[ (2x + 2)^2 = 4x^2 + 8x + 4 \]
- Right side:
\[ x(3x + 3) = 3x^2 + 3x \]

4. **Set the equation**:
\[ 4x^2 + 8x + 4 = 3x^2 + 3x \]

5. **Rearrange the equation**:
\[ 4x^2 - 3x^2 + 8x - 3x + 4 = 0 \]
This simplifies to:
\[ x^2 + 5x + 4 = 0 \]

6. **Factor the quadratic equation**:
\[ (x + 1)(x + 4) = 0 \]
Therefore, the solutions for \(x\) are:
\[ x = -1 \quad \text{or} \quad x = -4 \]

7. **Evaluate the terms for each value of \(x\)**:
- For \(x = -1\):
- First term: \(a = -1\)
- Second term: \(b = 2(-1) + 2 = 0\)
- Third term: \(c = 3(-1) + 3 = 0\)
- Since one of the terms is zero, they cannot be in G.P.

- For \(x = -4\):
- First term: \(a = -4\)
- Second term: \(b = 2(-4) + 2 = -6\)
- Third term: \(c = 3(-4) + 3 = -9\)

8. **Check if these terms are in G.P.**:
- The common ratio \(r\) can be calculated as:
\[ r = \frac{b}{a} = \frac{-6}{-4} = \frac{3}{2} \]
\[ r = \frac{c}{b} = \frac{-9}{-6} = \frac{3}{2} \]
- Since both ratios are equal, the terms are indeed in G.P.

9. **Find the fourth term**:
- The fourth term \(d\) can be calculated as:
\[ d = c \cdot r = -9 \cdot \frac{3}{2} = -\frac{27}{2} = -13.5 \]

### Final Answer:
The fourth term is \(-13.5\).