Let `alpha_1,beta_1` be the roots `x^2-6x+p=0a n d` `alpha_2,beta_2` be the roots `x^2-54 x+q=0dot` If `alpha_1,beta_1,alpha_2,beta_2` form an increasing G.P., then sum of the digits of the value of `(q-p)` is ___________.

Explanation

To solve the problem, we need to find the values of \( p \) and \( q \) based on the roots of the given quadratic equations and then compute \( q - p \). Finally, we will find the sum of the digits of \( q - p \).

### Step 1: Define the roots
Let the roots of the first equation \( x^2 - 6x + p = 0 \) be \( \alpha_1 \) and \( \beta_1 \).
Let the roots of the second equation \( x^2 - 54x + q = 0 \) be \( \alpha_2 \) and \( \beta_2 \).

### Step 2: Express roots in terms of a common ratio
Since \( \alpha_1, \beta_1, \alpha_2, \beta_2 \) form an increasing geometric progression (G.P.), we can express the roots as:
- \( \alpha_1 = a \)
- \( \beta_1 = ar \)
- \( \alpha_2 = ar^2 \)
- \( \beta_2 = ar^3 \)

where \( r \) is the common ratio.

### Step 3: Use Vieta's formulas
From Vieta's formulas:
1. For the first equation:
- \( \alpha_1 + \beta_1 = 6 \) implies \( a + ar = 6 \) or \( a(1 + r) = 6 \) (Equation 1)
- \( \alpha_1 \cdot \beta_1 = p \) implies \( a \cdot ar = p \) or \( a^2 r = p \) (Equation 2)

2. For the second equation:
- \( \alpha_2 + \beta_2 = 54 \) implies \( ar^2 + ar^3 = 54 \) or \( ar^2(1 + r) = 54 \) (Equation 3)
- \( \alpha_2 \cdot \beta_2 = q \) implies \( ar^2 \cdot ar^3 = q \) or \( a^2 r^5 = q \) (Equation 4)

### Step 4: Solve for \( r \)
From Equation 1, we can express \( a \):
\[ a = \frac{6}{1 + r} \]
Substituting \( a \) into Equation 2:
\[ p = \left(\frac{6}{1 + r}\right)^2 r = \frac{36r}{(1 + r)^2} \]

Now substituting \( a \) into Equation 3:
\[ ar^2(1 + r) = 54 \implies \frac{6r^2}{1 + r} (1 + r) = 54 \implies 6r^2 = 54 \implies r^2 = 9 \implies r = 3 \text{ (since it's an increasing G.P.)} \]

### Step 5: Find \( a \)
Substituting \( r = 3 \) back into the equation for \( a \):
\[ a = \frac{6}{1 + 3} = \frac{6}{4} = \frac{3}{2} \]

### Step 6: Calculate \( p \) and \( q \)
Now we can find \( p \) and \( q \):
1. Calculate \( p \):
\[ p = a^2 r = \left(\frac{3}{2}\right)^2 \cdot 3 = \frac{9}{4} \cdot 3 = \frac{27}{4} \]

2. Calculate \( q \):
\[ q = a^2 r^5 = \left(\frac{3}{2}\right)^2 \cdot 3^5 = \frac{9}{4} \cdot 243 = \frac{2187}{4} \]

### Step 7: Find \( q - p \)
Now we compute \( q - p \):
\[ q - p = \frac{2187}{4} - \frac{27}{4} = \frac{2187 - 27}{4} = \frac{2160}{4} = 540 \]

### Step 8: Sum of the digits of \( q - p \)
The value of \( q - p \) is \( 540 \). The sum of the digits is:
\[ 5 + 4 + 0 = 9 \]

### Final Answer:
The sum of the digits of the value of \( (q - p) \) is **9**.