Given:
The equation is ax^2 + bx + c = 0, where a, b, and c are in an arithmetic progression (A.P.).
The sum of the roots is 5 greater than the product of the roots.

To find:
The modulus of the harmonic mean of the roots.

Solution:

Step 1: Find the roots of the quadratic equation:
Let the roots of the quadratic equation be α and β.

Sum of the roots (α + β) = -b/a
Product of the roots (α * β) = c/a

Given that the sum of the roots is 5 greater than the product of the roots:
α + β = 5 + αβ

Substituting the values of the sum and product of the roots:
-b/a = 5 + c/a

Simplifying the equation:
b/a = -5 - c/a
b = -5a - c

Substituting the value of b in the quadratic equation:
ax^2 + (-5a - c)x + c = 0

Step 2: Determine the arithmetic progression:
Given that a, b, and c are in an arithmetic progression (A.P.), we have:
b - a = c - b

Simplifying the equation:
2b = a + c
a = 2b - c

Substituting the value of a in the quadratic equation:
(2b - c)x^2 + (-5a - c)x + c = 0

Step 3: Apply the harmonic mean formula:
The harmonic mean of two numbers is given by:
Harmonic Mean = 2/(1/α + 1/β)

Let the roots α and β be in the form (b - m) and (b + m), where m is the common difference of the arithmetic progression.

Substituting the values in the harmonic mean formula:
Harmonic Mean = 2/(1/(b - m) + 1/(b + m))

Simplifying the expression:
Harmonic Mean = 2/((2b)/(b^2 - m^2))
Harmonic Mean = (b^2 - m^2)/(2b)

Step 4: Calculate the modulus of the harmonic mean:
The modulus of the harmonic mean is given by the absolute value of the harmonic mean.

Taking the absolute value of the expression:
|Harmonic Mean| = |(b^2 - m^2)/(2b)|

Since we need to find the modulus of the harmonic mean, it doesn't matter if the roots are positive or negative.

Step 5: Substitute the value of a in terms of b and c:
Substituting the value of a in terms of b and c:
a = 2b - c

Step 6: Simplify the expression:
|Harmonic Mean| = |((2b - c)^2 - m^2)/(2b)|

Step 7: Apply the condition that the sum of the roots is 5 greater than the product of the roots:
α + β = 5 + αβ

Substituting the values of α and β:
(b - m) + (b + m) = 5 + (b -