If `x,2x+2,3x+3` are the first three terms of a GP, then what is its fourth term?

Options

A. `-27//2`
B. `27//2`
C. `-33//2`
D. `33//2`

Explanation

To find the fourth term of the geometric progression (GP) given the first three terms as \(x\), \(2x + 2\), and \(3x + 3\), we can follow these steps:

### Step 1: Identify the terms of the GP
The first three terms of the GP are:
- \(a = x\)
- \(b = 2x + 2\)
- \(c = 3x + 3\)

### Step 2: Use the property of GP
In a geometric progression, the square of the middle term is equal to the product of the other two terms. This can be expressed as:
\[ b^2 = ac \]
Substituting the values of \(a\), \(b\), and \(c\):
\[ (2x + 2)^2 = x(3x + 3) \]

### Step 3: Expand both sides
Now, we will expand both sides of the equation:
\[ (2x + 2)^2 = 4x^2 + 8x + 4 \]
\[ x(3x + 3) = 3x^2 + 3x \]

### Step 4: Set up the equation
Now we can set the expanded forms equal to each other:
\[ 4x^2 + 8x + 4 = 3x^2 + 3x \]

### Step 5: Rearrange the equation
Rearranging gives us:
\[ 4x^2 - 3x^2 + 8x - 3x + 4 = 0 \]
This simplifies to:
\[ x^2 + 5x + 4 = 0 \]

### Step 6: Factor the quadratic equation
Next, we can factor the quadratic equation:
\[ (x + 4)(x + 1) = 0 \]

### Step 7: Solve for \(x\)
Setting each factor to zero gives us:
\[ x + 4 = 0 \quad \Rightarrow \quad x = -4 \]
\[ x + 1 = 0 \quad \Rightarrow \quad x = -1 \]

### Step 8: Find the fourth term
Now we need to find the fourth term of the GP. The fourth term \(d\) can be found using the formula for the \(n\)-th term of a GP:
\[ d = ar^3 \]
where \(r\) is the common ratio given by \(r = \frac{b}{a} = \frac{2x + 2}{x}\).

### Step 9: Calculate \(r\) for both values of \(x\)
1. For \(x = -4\):
\[ r = \frac{2(-4) + 2}{-4} = \frac{-8 + 2}{-4} = \frac{-6}{-4} = \frac{3}{2} \]
Now, calculate the fourth term:
\[ d = x \cdot r^3 = -4 \cdot \left(\frac{3}{2}\right)^3 = -4 \cdot \frac{27}{8} = -\frac{108}{8} = -13.5 \]

2. For \(x = -1\):
\[ r = \frac{2(-1) + 2}{-1} = \frac{-2 + 2}{-1} = \frac{0}{-1} = 0 \]
The fourth term in this case would be:
\[ d = -1 \cdot 0^3 = 0 \]

### Conclusion
Thus, the fourth term can be either \(-13.5\) or \(0\) depending on the value of \(x\).