Given information:
- a, b, c are in arithmetic progression.
- a/2, b, c/4 are in geometric progression.
- b = 2.

Approach:
To find the value of (a^4 b^4 c^4), we need to express a, b, and c in terms of a common ratio and then substitute the given values.

Solution:

Arithmetic Progression:
Let's assume the common difference of the arithmetic progression is 'd'.

The terms of the arithmetic progression can be expressed as:
a = a, b = a + d, c = a + 2d

Geometric Progression:
Let's assume the common ratio of the geometric progression is 'r'.

The terms of the geometric progression can be expressed as:
a/2 = a, b = ar, c/4 = ar^2

From the information given, we know that b = 2. Substituting this value, we get:
a + d = 2

Substituting common ratio in terms of common difference:
Using the given information, we can express the common ratio in terms of the common difference.

From the geometric progression, we have:
b = ar
2 = a(a + d)

Simplifying the equation, we get:
a^2 + ad - 2 = 0

Using the quadratic formula, we can solve for 'a':
a = (-d ± √(d^2 + 8))/2

Solving for 'a':
Since 'a' is a positive value, we can ignore the negative root of the quadratic equation.

Using the positive root, we have:
a = (-d + √(d^2 + 8))/2

Value of 'd':
To find the value of 'd', we can substitute the value of 'a' into the equation a + d = 2.

(-d + √(d^2 + 8))/2 + d = 2

Simplifying the equation, we get:
√(d^2 + 8) = 4 - d

Squaring both sides of the equation, we have:
d^2 + 8 = 16 - 8d + d^2

Simplifying the equation, we get:
8d = 8

Therefore, d = 1.

Calculating a, b, and c:
Using the value of 'd', we can calculate the values of a, b, and c.

a = (-1 + √(1 + 8))/2
a = (-1 + 3)/2
a = 1

b = a + d
b = 1 + 1
b = 2

c = a + 2d
c = 1 + 2(1)
c = 3

Calculating the value of (a^4 b^4 c^4):
Substituting the values of a, b, and c, we have:
(a^4 b^4 c^4) = (1^4 2^4 3^4)
(a^4 b^4 c^4) = (1 16 81)
(a