To determine the conditions for the roots of the equation x^3 + bx^2 + cx - 1 = 0 to form an increasing geometric progression (G.P.), we need to analyze the equation and its properties.

Analysis of the Equation:
The equation x^3 + bx^2 + cx - 1 = 0 can be written as:
x^3 + bx^2 + cx = 1

Key Points:
1. Sum of Roots:
The sum of the roots of the equation is given by the coefficient of x^2 with a negative sign, i.e., -b. Therefore, the sum of the roots is -b.

2. Product of Roots:
The product of the roots is given by the constant term, i.e., -1.

Conditions for Roots to Form an Increasing G.P.:
For the roots of the equation to form an increasing G.P., the following conditions must be satisfied:

A. Roots in an Increasing Order:
The roots should be arranged in an increasing order.

B. Common Ratio of the G.P.:
The common ratio (r) of the G.P. can be obtained by dividing the second root by the first root. Let the roots be a, ar, and ar^2. Therefore, the common ratio (r) can be expressed as:
r = (ar)/(a) = ar/a = r^2

C. Sum of Roots in G.P. Form:
The sum of the roots should be equal to the sum of the terms of a G.P. with the first term as a and the common ratio as r. The sum of the terms of a G.P. can be expressed as:
Sum = a(1 - r^n)/(1 - r)
where n is the number of terms in the G.P.

D. Product of Roots in G.P. Form:
The product of the roots should be equal to the product of the terms of a G.P. with the first term as a and the common ratio as r. The product of the terms of a G.P. can be expressed as:
Product = a^n

Combining the Conditions:
To satisfy all the conditions mentioned above, the following equations need to be satisfied simultaneously:

1. Roots in an Increasing Order: No specific equation required.

2. Common Ratio of the G.P.:
r = r^2
This equation implies that r = 0 or r = 1.

3. Sum of Roots in G.P. Form:
Sum = a(1 - r^n)/(1 - r) = -b
This equation relates a, r, and b.

4. Product of Roots in G.P. Form:
Product = a^n = -1
This equation relates a and n.

Based on the above analysis and conditions, we can conclude the following:

A. Roots can be arranged in an increasing order if:
No specific equation required.

B. Possible values for the common ratio (r) are:
r = 0 or r = 1

C. The sum of roots (a(1 - r^n)/(1 - r)) can be equal to -b if:
b, c ∈ (-∞, -3)

D. The product of roots (a^n) can be equal to -1 if:
One of the roots is 1

Therefore, the correct answer is option 'A, B, C, D'.

Note: The above explanation and conditions are applicable for the