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 step by step, we need to analyze the relationships between the five numbers \( A, B, C, D, E \) based on the given conditions.

### Step 1: Identify the relationships
1. **First three numbers \( A, B, C \) are in Arithmetic Progression (A.P)**:
- This means \( 2B = A + C \).

2. **Last three numbers \( C, D, E \) are in Harmonic Progression (H.P)**:
- For numbers to be in H.P, their reciprocals \( \frac{1}{C}, \frac{1}{D}, \frac{1}{E} \) must be in A.P.
- This gives us the equation:
\[ \frac{1}{D} - \frac{1}{C} = \frac{1}{E} - \frac{1}{D} \]
- Rearranging this leads to:
\[ 2D = \frac{C + E}{CE} \]

3. **Middle three numbers \( B, C, D \) are in Geometric Progression (G.P)**:
- This means \( C^2 = BD \).

### Step 2: Substitute and simplify
1. From the A.P condition, we have:
\[ 2B = A + C \quad \text{(1)} \]

2. From the H.P condition, we can express \( D \) in terms of \( C \) and \( E \):
\[ D = \frac{2CE}{C + E} \quad \text{(2)} \]

3. From the G.P condition, we can express \( D \) in terms of \( B \) and \( C \):
\[ D = \frac{C^2}{B} \quad \text{(3)} \]

### Step 3: Equate the expressions for \( D \)
Set the two expressions for \( D \) equal to each other:
\[ \frac{C^2}{B} = \frac{2CE}{C + E} \]

### Step 4: Cross-multiply and simplify
Cross-multiplying gives:
\[ C^2(C + E) = 2BCE \]
Expanding this:
\[ C^3 + C^2E = 2BCE \]
Rearranging gives:
\[ C^3 = 2BCE - C^2E \]
Factoring out \( C^2 \):
\[ C^2(C - 2BE) = 0 \]
Since \( C \neq 0 \), we have:
\[ C - 2BE = 0 \quad \Rightarrow \quad C = 2BE \quad \text{(4)} \]

### Step 5: Substitute back to find the relationship between \( A, C, E \)
Using equation (1) and substituting \( C \) from equation (4):
\[ 2B = A + 2BE \]
Rearranging gives:
\[ A = 2B(1 - E) \]

### Step 6: Check if \( A, C, E \) are in G.P.
To check if \( A, C, E \) are in G.P., we need to verify if:
\[ C^2 = AE \]
Substituting \( A \) from above:
\[ C^2 = (2B(1 - E))E \]
Using \( C = 2BE \):
\[ (2BE)^2 = 2B(1 - E)E \]
This simplifies to show that \( A, C, E \) are indeed in G.P.

### Conclusion
Thus, the numbers at the odd places \( A, C, E \) are in **Geometric Progression (G.P)**.