If a, 2a+2, 3a+3 are in GP, then what is the fourth term of the GP ?
(a)`-13.5`
(b)`13.5`
(c)`-27`
(d)`27`

Options

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

Answer

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

### Step 1: Identify the terms
The three terms of the GP are:
- First term: \( a \)
- Second term: \( 2a + 2 \)
- Third term: \( 3a + 3 \)

### Step 2: Use the property of GP
In a GP, the square of the second term is equal to the product of the first and third terms. This can be expressed as:
\[ (2a + 2)^2 = a \cdot (3a + 3) \]

### Step 3: Expand both sides
Expanding the left side:
\[ (2a + 2)^2 = 4a^2 + 8a + 4 \]
Expanding the right side:
\[ a \cdot (3a + 3) = 3a^2 + 3a \]

### Step 4: Set the equation
Now we set the two expansions equal to each other:
\[ 4a^2 + 8a + 4 = 3a^2 + 3a \]

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

### Step 6: Factor the quadratic equation
Now we factor the quadratic:
\[ (a + 4)(a + 1) = 0 \]
This gives us two possible solutions:
\[ a + 4 = 0 \quad \Rightarrow \quad a = -4 \]
\[ a + 1 = 0 \quad \Rightarrow \quad a = -1 \]

### Step 7: Find the terms for each value of \( a \)
1. For \( a = -4 \):
- First term: \( -4 \)
- Second term: \( 2(-4) + 2 = -8 + 2 = -6 \)
- Third term: \( 3(-4) + 3 = -12 + 3 = -9 \)

2. For \( a = -1 \):
- First term: \( -1 \)
- Second term: \( 2(-1) + 2 = -2 + 2 = 0 \)
- Third term: \( 3(-1) + 3 = -3 + 3 = 0 \)

Since the second and third terms must be non-zero for a valid GP, we will use \( a = -4 \).

### Step 8: Calculate the common ratio \( r \)
The common ratio \( r \) can be calculated as:
\[ r = \frac{\text{Second term}}{\text{First term}} = \frac{-6}{-4} = \frac{3}{2} \]

### Step 9: Calculate the fourth term
The fourth term can be calculated using:
\[ \text{Fourth term} = \text{Third term} \times r = -9 \times \frac{3}{2} = -\frac{27}{2} = -13.5 \]

### Conclusion
Thus, the fourth term of the GP is:
\[ \boxed{-13.5} \]