The values of x for which the numbers `(-2)/7,x,7/2` are in G.P is

Options

A. 0
B. `pm1`
C. `+1`
D. `-1`

Answer

To determine the values of \( x \) for which the numbers \( -\frac{2}{7}, x, \frac{7}{2} \) are in geometric progression (G.P.), we can use the property of G.P. that states if \( a, b, c \) are in G.P., then \( b^2 = a \cdot c \).

### Step-by-Step Solution:

1. **Identify the terms in G.P.**:
Let \( a = -\frac{2}{7} \), \( b = x \), and \( c = \frac{7}{2} \).

2. **Apply the G.P. condition**:
According to the G.P. condition, we have:
\[ b^2 = a \cdot c \]
Substituting the values:
\[ x^2 = \left(-\frac{2}{7}\right) \cdot \left(\frac{7}{2}\right) \]

3. **Calculate the right-hand side**:
Simplifying the right-hand side:
\[ x^2 = -\frac{2}{7} \cdot \frac{7}{2} = -1 \]

4. **Solve for \( x \)**:
Now we have:
\[ x^2 = -1 \]
Taking the square root of both sides gives:
\[ x = \pm i \]
where \( i \) is the imaginary unit.

5. **Final answer**:
Therefore, the values of \( x \) for which the numbers \( -\frac{2}{7}, x, \frac{7}{2} \) are in G.P. are:
\[ x = i \quad \text{and} \quad x = -i \]