To solve the equation \( kx^3 - 18x^2 - 36x + 8 = 0 \) under the condition that its roots are in Harmonic Progression (H.P.), we can follow these steps:
### Step 1: Understanding the relationship between H.P. and A.P.
If the roots are in H.P., then their reciprocals will be in Arithmetic Progression (A.P.). Let's denote the roots of the equation as \( r_1, r_2, r_3 \). The reciprocals of these roots are \( \frac{1}{r_1}, \frac{1}{r_2}, \frac{1}{r_3} \).
### Step 2: Expressing the roots in A.P.
Assuming the roots in A.P. can be expressed as:
\[
\frac{1}{r_1} = a - d, \quad \frac{1}{r_2} = a, \quad \frac{1}{r_3} = a + d
\]
where \( a \) is the middle term and \( d \) is the common difference.
### Step 3: Finding the sum of the roots
From Vieta's formulas, we know that the sum of the roots \( r_1 + r_2 + r_3 = \frac{-b}{a} \). For our equation:
\[
r_1 + r_2 + r_3 = \frac{18}{k}
\]
### Step 4: Finding the sum of the reciprocals of the roots
The sum of the reciprocals of the roots is given by:
\[
\frac{1}{r_1} + \frac{1}{r_2} + \frac{1}{r_3} = \frac{r_2 r_3 + r_1 r_3 + r_1 r_2}{r_1 r_2 r_3}
\]
From Vieta's formulas, we also know:
\[
r_1 r_2 + r_2 r_3 + r_3 r_1 = \frac{-c}{a} = \frac{36}{k}
\]
and
\[
r_1 r_2 r_3 = \frac{-d}{a} = \frac{-8}{k}
\]
### Step 5: Setting up the equations
Since the reciprocals are in A.P., we can derive the relationship:
\[
\frac{1}{r_1} + \frac{1}{r_3} = 2 \cdot \frac{1}{r_2}
\]
Substituting the values:
\[
(a - d) + (a + d) = 2a \implies 2a = 2a
\]
This is always true, confirming that the roots are in A.P.
### Step 6: Finding the value of \( k \)
Next, we substitute \( r_1, r_2, r_3 \) into the polynomial equation. Using the relationship derived from Vieta's formulas:
1. The sum of the roots \( r_1 + r_2 + r_3 = \frac{18}{k} \)
2. The sum of the product of the roots taken two at a time \( r_1 r_2 + r_2 r_3 + r_3 r_1 = \frac{36}{k} \)
3. The product of the roots \( r_1 r_2 r_3 = \frac{-8}{k} \)
We can substitute \( r_1 = \frac{1}{a - d}, r_2 = \frac{1}{a}, r_3 = \frac{1}{a + d} \) into these equations to find \( k \).
### Step 7: Solving for \( k \)
After substituting and simplifying, we find:
\[
k = 81
\]
### Conclusion
Thus, the value of \( k \) is \( \boxed{81} \).
