If x, 2x + 2, 3x + 3,….are in G.P., then the fourth term is

Options

A. 27
B. -27
C. 135
D. -13.5

Answer

To solve the problem, we need to determine the fourth term of the sequence \( x, 2x + 2, 3x + 3 \) given that these terms are in a geometric progression (G.P.).

### Step-by-step Solution:

1. **Identify the terms**:
The first term \( T_1 = x \), the second term \( T_2 = 2x + 2 \), and the third term \( T_3 = 3x + 3 \).

2. **Use the property of G.P.**:
For three terms to be in G.P., the ratio of the second term to the first term must be equal to the ratio of the third term to the second term. This gives us the equation:
\[ \frac{T_2}{T_1} = \frac{T_3}{T_2} \]

3. **Set up the equation**:
Substituting the terms into the equation:
\[ \frac{2x + 2}{x} = \frac{3x + 3}{2x + 2} \]

4. **Cross-multiply**:
Cross-multiplying gives:
\[ (2x + 2)^2 = x(3x + 3) \]

5. **Expand both sides**:
Expanding the left side:
\[ 4x^2 + 8x + 4 \]
Expanding the right side:
\[ 3x^2 + 3x \]

6. **Set the equation to zero**:
Setting both sides equal gives:
\[ 4x^2 + 8x + 4 = 3x^2 + 3x \]
Rearranging gives:
\[ 4x^2 - 3x^2 + 8x - 3x + 4 = 0 \]
Simplifying leads to:
\[ x^2 + 5x + 4 = 0 \]

7. **Factor the quadratic**:
Factoring the quadratic equation:
\[ (x + 4)(x + 1) = 0 \]
This gives us the solutions:
\[ x = -4 \quad \text{or} \quad x = -1 \]

8. **Find the common ratio**:
We can use either value of \( x \) to find the common ratio \( r \). Let's use \( x = -1 \):
\[ T_1 = -1, \quad T_2 = 0, \quad T_3 = 2 \]
The common ratio \( r \) is:
\[ r = \frac{T_2}{T_1} = \frac{0}{-1} = 0 \quad \text{(not valid)} \]
Now using \( x = -4 \):
\[ T_1 = -4, \quad T_2 = -6, \quad T_3 = -9 \]
The common ratio \( r \) is:
\[ r = \frac{T_2}{T_1} = \frac{-6}{-4} = \frac{3}{2} \]

9. **Calculate the fourth term**:
The fourth term \( T_4 \) can be calculated using the formula:
\[ T_4 = T_3 \cdot r = -9 \cdot \frac{3}{2} = -\frac{27}{2} = -13.5 \]

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