If p,q,r are in Geometric progression, then which is true among the following

Options

A. `q=(p+r)/2`
B. `p^2=qr`
C. `q=sqrt(pr)`
D. `p/r=r/q`

Answer

To determine which statement is true when \( p, q, r \) are in geometric progression (GP), we start by recalling the definition of a geometric progression. In a GP, the square of the middle term is equal to the product of the other two terms.

### Step-by-step Solution:

1. **Understanding Geometric Progression**:
If \( p, q, r \) are in GP, then by definition:
\[ q^2 = p \cdot r \]

2. **Rearranging the Equation**:
To find \( q \), we can take the square root of both sides:
\[ q = \sqrt{p \cdot r} \]

3. **Identifying the Correct Option**:
Now, we need to compare this result with the provided options:
- Option A: \( q = \frac{p + r}{2} \) (This represents the arithmetic mean, not GP)
- Option B: \( p^2 = q \cdot r \) (This is not a correct representation of GP)
- Option C: \( q = \sqrt{p \cdot r} \) (This matches our derived equation)
- Option D: \( \frac{p}{r} = \frac{r}{q} \) (This is a different relationship)

From our analysis, we find that **Option C** is correct.

### Final Answer:
The correct statement is:
\[ q = \sqrt{p \cdot r} \]