a,b,c,d,e are five numbers in which the first three are in A.P and the last three are in H.P .If the three numbers in the middle are in G.P then the numbers at the odd places are in

Options

A. A.P
B. G.P
C. H.P
D. none of these

Answer

To solve the problem, we need to analyze the relationships between the numbers \( a, b, c, d, e \) based on the given conditions.

### Step-by-Step Solution:

1. **Understanding the Conditions**:
- The first three numbers \( a, b, c \) are in Arithmetic Progression (A.P.).
- The last three numbers \( c, d, e \) are in Harmonic Progression (H.P.).
- The middle three numbers \( b, c, d \) are in Geometric Progression (G.P.).

2. **Using the A.P. Condition**:
Since \( a, b, c \) are in A.P., we have:
\[ 2b = a + c \quad \text{(1)} \]

3. **Using the G.P. Condition**:
Since \( b, c, d \) are in G.P., we have:
\[ c^2 = b \cdot d \quad \text{(2)} \]

4. **Using the H.P. Condition**:
Since \( c, d, e \) are in H.P., we can express this condition as:
\[ \frac{1}{c}, \frac{1}{d}, \frac{1}{e} \text{ are in A.P.} \]
This gives us:
\[ 2\frac{1}{d} = \frac{1}{c} + \frac{1}{e} \]
Multiplying through by \( cde \) gives:
\[ 2ce = e + c \quad \text{(3)} \]

5. **Rearranging Equation (3)**:
Rearranging equation (3) gives:
\[ 2ce = e + c \implies 2ce - c = e \implies c(2e - 1) = e \quad \text{(4)} \]

6. **Substituting for \( d \)**:
From equation (2), we can express \( d \):
\[ d = \frac{c^2}{b} \quad \text{(5)} \]

7. **Substituting \( d \) in the H.P. Condition**:
Substitute \( d \) from equation (5) into equation (3):
\[ 2 \cdot \frac{c^2}{b} = 2ce - c \implies 2c^2 = b(2ce - c) \]

8. **Finding the Relationship**:
From the equations derived, we can analyze the relationship between \( a, c, e \). From equation (1), we have:
\[ a = 2b - c \]
Using \( c(2e - 1) = e \) from (4), we can express \( e \) in terms of \( c \).

9. **Conclusion**:
We find that:
\[ c^2 = ae \quad \text{(from the derived relationships)} \]
This implies that \( a, c, e \) are in Geometric Progression (G.P.).

### Final Result:
The numbers at the odd places \( a, c, e \) are in G.P.