Given a,b,c are in A.P.,b,c,d are in G.P and c,d,e are in H.P .If a=2 and e=18 , then the sum of all possible value of c is ________.

Explanation

To solve the problem step by step, we start with the given conditions and equations based on the relationships between the terms in Arithmetic Progression (A.P.), Geometric Progression (G.P.), and Harmonic Progression (H.P.).

### Step 1: Understand the relationships
Given:
- \( a, b, c \) are in A.P.
- \( b, c, d \) are in G.P.
- \( c, d, e \) are in H.P.

From the properties of A.P., we know that:
\[ b - a = c - b \implies 2b = a + c \implies b = \frac{a + c}{2} \]
This is our **Equation (1)**.

### Step 2: Use the G.P. condition
For \( b, c, d \) in G.P., we have:
\[ \frac{c}{b} = \frac{d}{c} \implies c^2 = bd \]
This is our **Equation (2)**.

### Step 3: Use the H.P. condition
For \( c, d, e \) in H.P., we have:
\[ d = \frac{2ce}{c + e} \]
This is our **Equation (3)**.

### Step 4: Substitute values for \( a \) and \( e \)
We know \( a = 2 \) and \( e = 18 \). Substitute these values into our equations.

From **Equation (1)**:
\[ b = \frac{2 + c}{2} \]

From **Equation (3)**:
\[ d = \frac{2c \cdot 18}{c + 18} = \frac{36c}{c + 18} \]

### Step 5: Substitute \( b \) and \( d \) into Equation (2)
Now substitute \( b \) and \( d \) into **Equation (2)**:
\[ c^2 = bd = \left(\frac{2 + c}{2}\right) \left(\frac{36c}{c + 18}\right) \]

### Step 6: Simplify the equation
Cross-multiply to eliminate the fractions:
\[ c^2(c + 18) = (2 + c)(18c) \]
Expanding both sides:
\[ c^3 + 18c^2 = 36c + 18c^2 \]
Subtract \( 18c^2 \) from both sides:
\[ c^3 = 36c \]

### Step 7: Factor the equation
Rearranging gives:
\[ c^3 - 36c = 0 \]
Factoring out \( c \):
\[ c(c^2 - 36) = 0 \]
This gives:
\[ c(c - 6)(c + 6) = 0 \]

### Step 8: Solve for \( c \)
The solutions are:
\[ c = 0, \quad c = 6, \quad c = -6 \]

### Step 9: Find the sum of all possible values of \( c \)
Now, we sum all possible values of \( c \):
\[ 0 + 6 + (-6) = 0 \]

### Final Answer
The sum of all possible values of \( c \) is:
\[ \boxed{0} \]