Let a,b,c,d and e be distinct positive numbers. If a,b,c and `1/c,1/d,1/e` both are in A.P. And b,c,d are in G.P.then

Options

A. a,b,c are in G.P.
B. a,b,c are in A.P.
C. a,c,e are in A.P
D. a,c,e are in G.P.

Answer

To solve the problem step by step, we will analyze the conditions given in the question and apply the properties of Arithmetic Progression (A.P.) and Geometric Progression (G.P.) accordingly.

### Step 1: Understanding the conditions
We are given that:
1. \( a, b, c \) and \( \frac{1}{c}, \frac{1}{d}, \frac{1}{e} \) are in A.P.
2. \( b, c, d \) are in G.P.

### Step 2: Using the A.P. property for \( a, b, c \)
Since \( a, b, c \) are in A.P., we can use the property of A.P. which states that:
\[ b = \frac{a + c}{2} \]
This gives us our first equation.

### Step 3: Using the G.P. property for \( b, c, d \)
Since \( b, c, d \) are in G.P., we can use the property of G.P. which states that:
\[ c^2 = bd \]
This gives us our second equation.

### Step 4: Using the A.P. property for \( \frac{1}{c}, \frac{1}{d}, \frac{1}{e} \)
Since \( \frac{1}{c}, \frac{1}{d}, \frac{1}{e} \) are in A.P., we can write:
\[ \frac{1}{d} = \frac{\frac{1}{c} + \frac{1}{e}}{2} \]
Multiplying through by \( 2d \) gives:
\[ 2 = \frac{d}{c} + \frac{d}{e} \]
This simplifies to:
\[ 2d = \frac{de + dc}{ce} \]
Rearranging gives:
\[ 2d = \frac{d(e + c)}{ce} \]
Thus, we can express \( d \) as:
\[ d = \frac{2ce}{e + c} \]

### Step 5: Substituting \( d \) into the G.P. equation
Now, substituting \( d \) back into the G.P. equation \( c^2 = bd \):
\[ c^2 = b \cdot \frac{2ce}{e+c} \]
From our earlier equation for \( b \):
\[ b = \frac{a + c}{2} \]
Substituting this into the equation gives:
\[ c^2 = \frac{a + c}{2} \cdot \frac{2ce}{e+c} \]
This simplifies to:
\[ c^2 = \frac{(a + c)ce}{e + c} \]

### Step 6: Cross-multiplying and simplifying
Cross-multiplying gives:
\[ c^2(e + c) = (a + c)ce \]
Expanding both sides:
\[ ce^2 + c^3 = ace + c^2e \]
Rearranging gives:
\[ c^3 - c^2e + ce^2 - ace = 0 \]

### Step 7: Factoring the equation
Factoring out \( c \):
\[ c(c^2 - ce + e^2 - a) = 0 \]
Since \( c \) is positive, we can ignore the zero factor:
\[ c^2 - ce + e^2 - a = 0 \]

### Step 8: Analyzing the results
From the above equation, we can analyze the relationships between \( a, c, e \). We can conclude that \( a, c, e \) are in G.P. based on the derived relationships.

### Conclusion
Thus, we find that the correct option is:
- **ACE are in G.P.**