If the positive integers a,b,c and d are in AP, then the numbers abc, abd,acd and bcd are in

Options

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

Answer

To solve the problem, we need to determine the relationship between the products of the numbers \( abc, abd, acd, \) and \( bcd \) when \( a, b, c, \) and \( d \) are in arithmetic progression (AP).

### Step-by-Step Solution:

1. **Understanding Arithmetic Progression (AP)**:
- If \( a, b, c, d \) are in AP, then there exists a common difference \( d \) such that:
\[ b - a = c - b = d - c \]
- We can express \( b, c, d \) in terms of \( a \) and the common difference \( d \):
\[ b = a + k, \quad c = a + 2k, \quad d = a + 3k \]
- Here, \( k \) is the common difference.

2. **Calculating the Products**:
- We need to find the products:
\[ abc = a \cdot b \cdot c = a \cdot (a + k) \cdot (a + 2k) \]
\[ abd = a \cdot b \cdot d = a \cdot (a + k) \cdot (a + 3k) \]
\[ acd = a \cdot c \cdot d = a \cdot (a + 2k) \cdot (a + 3k) \]
\[ bcd = b \cdot c \cdot d = (a + k) \cdot (a + 2k) \cdot (a + 3k) \]

3. **Identifying the Relationship**:
- We need to check if these products are in any specific type of progression (harmonic, arithmetic, or geometric).
- Recall that if \( x_1, x_2, x_3, x_4 \) are in harmonic progression, then the reciprocals \( \frac{1}{x_1}, \frac{1}{x_2}, \frac{1}{x_3}, \frac{1}{x_4} \) must be in arithmetic progression.

4. **Reciprocals of the Products**:
- Calculate the reciprocals:
\[ \frac{1}{abc}, \quad \frac{1}{abd}, \quad \frac{1}{acd}, \quad \frac{1}{bcd} \]
- These can be expressed as:
\[ \frac{1}{abc} = \frac{1}{a \cdot (a + k) \cdot (a + 2k)}, \quad \frac{1}{abd} = \frac{1}{a \cdot (a + k) \cdot (a + 3k)} \]
\[ \frac{1}{acd} = \frac{1}{a \cdot (a + 2k) \cdot (a + 3k)}, \quad \frac{1}{bcd} = \frac{1}{(a + k) \cdot (a + 2k) \cdot (a + 3k)} \]

5. **Checking for Harmonic Progression**:
- To check if these reciprocals are in arithmetic progression, we can analyze the structure of the terms.
- If we denote \( A = a, B = a + k, C = a + 2k, D = a + 3k \), we can see that:
\[ \frac{1}{A}, \frac{1}{B}, \frac{1}{C}, \frac{1}{D} \]
- The relationship between these terms will show that they are in harmonic progression.

6. **Conclusion**:
- Since the reciprocals of the products \( abc, abd, acd, bcd \) are in arithmetic progression, we conclude that the products themselves are in harmonic progression.

### Final Answer:
The numbers \( abc, abd, acd, \) and \( bcd \) are in **harmonic progression**.