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 possible value of 'c' can be :

Options

A. 9
B. `-6`
C. 6
D. `-9`

Answer

To solve the problem step-by-step, we need to use the properties of arithmetic progression (A.P.), geometric progression (G.P.), and harmonic progression (H.P.) as given in the question.

### Step 1: Identify the relationships based on the progressions

1. **A.P. (Arithmetic Progression)**: For \( a, b, c \) in A.P., we have:
\[ b = \frac{a + c}{2} \]

2. **G.P. (Geometric Progression)**: For \( b, c, d \) in G.P., we have:
\[ c^2 = b \cdot d \]

3. **H.P. (Harmonic Progression)**: For \( c, d, e \) in H.P., we have:
\[ d = \frac{2ce}{c + e} \]

### Step 2: Substitute known values

We are given \( a = 2 \) and \( e = 18 \).

### Step 3: Substitute \( a \) into the A.P. equation

Using the A.P. equation:
\[ b = \frac{2 + c}{2} \]

### Step 4: Substitute \( b \) into the G.P. equation

Now substitute \( b \) into the G.P. equation:
\[ c^2 = \left(\frac{2 + c}{2}\right) \cdot d \]

### Step 5: Substitute \( d \) from the H.P. equation

Now substitute \( d \) from the H.P. equation into the G.P. equation:
\[ c^2 = \left(\frac{2 + c}{2}\right) \cdot \left(\frac{2ce}{c + e}\right) \]

Substituting \( e = 18 \):
\[ c^2 = \left(\frac{2 + c}{2}\right) \cdot \left(\frac{2c \cdot 18}{c + 18}\right) \]

### Step 6: Simplify the equation

1. Cancel the 2's:
\[ c^2 = \frac{(2 + c) \cdot (36c)}{c + 18} \]

2. Cross-multiply:
\[ c^2(c + 18) = 36c(2 + c) \]

3. Expand both sides:
\[ c^3 + 18c^2 = 72c + 36c^2 \]

4. Rearranging gives:
\[ c^3 + 18c^2 - 36c^2 - 72c = 0 \]
\[ c^3 - 18c^2 - 72c = 0 \]

### Step 7: Factor the equation

Factor out \( c \):
\[ c(c^2 - 18c - 72) = 0 \]

### Step 8: Solve the quadratic equation

Now we solve the quadratic equation \( c^2 - 18c - 72 = 0 \) using the quadratic formula:
\[ c = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 1, b = -18, c = -72 \):
\[ c = \frac{18 \pm \sqrt{(-18)^2 - 4 \cdot 1 \cdot (-72)}}{2 \cdot 1} \]
\[ c = \frac{18 \pm \sqrt{324 + 288}}{2} \]
\[ c = \frac{18 \pm \sqrt{612}}{2} \]
\[ c = \frac{18 \pm 6\sqrt{17}}{2} \]
\[ c = 9 \pm 3\sqrt{17} \]

### Step 9: Possible values of \( c \)

Thus, the possible values of \( c \) are:
\[ c = 9 + 3\sqrt{17} \quad \text{or} \quad c = 9 - 3\sqrt{17} \]

### Final Answer

The possible values of \( c \) can be:
1. \( 9 + 3\sqrt{17} \)
2. \( 9 - 3\sqrt{17} \)

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