If a, b and c be three distinct real number in G.P. and `a + b + c = xb`, then x cannot be

Options

A. 4
B. `-3`
C. `-2`
D. 2

Answer

To solve the given problem, we need to analyze the condition that \( a, b, c \) are in geometric progression (G.P.) and that \( a + b + c = xb \).

### Step-by-Step Solution:

1. **Assume the values in G.P.**:
Let \( a = A \), \( b = AR \), and \( c = AR^2 \), where \( R \) is the common ratio. Since \( a, b, c \) are distinct, \( R \) must not be equal to 1.

2. **Set up the equation**:
According to the problem, we have:
\[ a + b + c = xb \]
Substituting the values of \( a, b, c \):
\[ A + AR + AR^2 = x(AR) \]

3. **Factor out \( A \)**:
Since \( A \neq 0 \) (as \( a, b, c \) are distinct real numbers), we can divide the entire equation by \( A \):
\[ 1 + R + R^2 = xR \]

4. **Rearranging the equation**:
Rearranging gives us:
\[ R^2 - xR + (1 + R) = 0 \]

5. **Form a quadratic equation**:
This is a quadratic equation in \( R \):
\[ R^2 - xR + (1 + R) = 0 \]
Rearranging further, we get:
\[ R^2 + (1 - x)R + 1 = 0 \]

6. **Finding the discriminant**:
For \( R \) to have real roots, the discriminant of this quadratic must be non-negative:
\[ D = (1 - x)^2 - 4(1) \]
Simplifying, we find:
\[ D = (1 - x)^2 - 4 \]

7. **Set the discriminant greater than or equal to zero**:
We need:
\[ (1 - x)^2 - 4 \geq 0 \]
This simplifies to:
\[ (1 - x - 2)(1 - x + 2) \geq 0 \]
or:
\[ (-x - 1)(-x + 3) \geq 0 \]

8. **Finding the critical points**:
The critical points from the factors are \( x = -1 \) and \( x = 3 \).

9. **Testing intervals**:
We test the intervals determined by these critical points:
- For \( x < -1 \): Both factors are positive, so the product is positive.
- For \( -1 < x < 3 \): One factor is negative and the other is positive, so the product is negative.
- For \( x > 3 \): Both factors are negative, so the product is positive.

10. **Conclusion**:
The inequality holds for \( x \leq -1 \) or \( x \geq 3 \). Therefore, \( x \) cannot be in the interval \( (-1, 3) \).

### Final Answer:
Thus, \( x \) cannot be in the interval:
\[ \boxed{(-1, 3)} \]