`"If "-(2)/(7),x,-(7)/(2)" are in G.P. Find the value of x."`

Explanation

To solve the problem, we need to find the value of \( x \) such that the numbers \( -\frac{2}{7}, x, -\frac{7}{2} \) are in geometric progression (G.P.).

### Step-by-step Solution:

1. **Understanding the terms in G.P.:**
In a geometric progression, the ratio of consecutive terms is constant. Therefore, we can set up the relationship:
\[ \frac{a_2}{a_1} = \frac{a_3}{a_2} \]
where \( a_1 = -\frac{2}{7} \), \( a_2 = x \), and \( a_3 = -\frac{7}{2} \).

2. **Substituting the values:**
Plugging in the values into the equation gives:
\[ \frac{x}{-\frac{2}{7}} = \frac{-\frac{7}{2}}{x} \]

3. **Cross-multiplying:**
To eliminate the fractions, we cross-multiply:
\[ x \cdot x = -\frac{7}{2} \cdot -\frac{2}{7} \]
This simplifies to:
\[ x^2 = \frac{7 \cdot 2}{2 \cdot 7} \]

4. **Simplifying the right side:**
The right side simplifies as follows:
\[ x^2 = \frac{14}{14} = 1 \]

5. **Taking the square root:**
To find \( x \), we take the square root of both sides:
\[ x = \pm 1 \]

### Final Answer:
Thus, the value of \( x \) is \( 1 \) or \( -1 \).

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