If p,q,r are in one geometric progression and a,b,c are in another geometric progression, then ap, bq, cr are in

Options

A. AP
B. GP
C. HP
D. None of these

Answer

To solve the problem, we need to establish the relationship between the terms given in the question. We know that \( p, q, r \) are in a geometric progression (GP) and \( a, b, c \) are also in a GP. We need to determine the nature of the terms \( ap, bq, cr \).

### Step-by-Step Solution:

1. **Understanding Geometric Progression**:
- A sequence is in geometric progression if the ratio of consecutive terms is constant.
- For \( p, q, r \) to be in GP, we can express them as:
\[ q = pr^{1/2} \quad \text{(where \( r \) is the common ratio)} \]
- Similarly, for \( a, b, c \):
\[ b = a \cdot k \quad \text{and} \quad c = a \cdot k^2 \quad \text{(where \( k \) is the common ratio)} \]

2. **Expressing the Terms**:
- We can express \( ap, bq, cr \) in terms of \( a, p, k \):
\[ ap = a \cdot p \]
\[ bq = (a \cdot k) \cdot q \]
\[ cr = (a \cdot k^2) \cdot r \]

3. **Substituting \( q \) and \( r \)**:
- Substitute \( q \) and \( r \) in the expressions:
\[ bq = (a \cdot k) \cdot (pr^{1/2}) = apk \cdot r^{1/2} \]
\[ cr = (a \cdot k^2) \cdot r \]

4. **Combining the Terms**:
- Now we have:
\[ ap, \quad apk \cdot r^{1/2}, \quad a \cdot k^2 \cdot r \]
- We can express them in a common form:
\[ ap = ap, \quad bq = ap \cdot k \cdot r^{1/2}, \quad cr = ap \cdot k^2 \cdot r \]

5. **Finding the Ratios**:
- Now we check the ratios:
\[ \frac{bq}{ap} = k \cdot r^{1/2} \]
\[ \frac{cr}{bq} = k \cdot r \]

6. **Conclusion**:
- Since the ratios \( \frac{bq}{ap} \) and \( \frac{cr}{bq} \) are constant, it follows that \( ap, bq, cr \) are also in geometric progression.

### Final Answer:
Thus, \( ap, bq, cr \) are in geometric progression.

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