Common difference= -4/3 or -3/4. the terms will be (4/3,-1,3/4)or (3/4,-1'4/3)

Understanding the Problem
To find the terms of a geometric progression (G.P.) given the sum and product of the first three terms, we denote the terms as a, ar, and ar^2, where 'a' is the first term and 'r' is the common ratio.
Given Conditions
- The sum of the first three terms is:
a + ar + ar^2 = 13/12
- The product of the first three terms is:
a * ar * ar^2 = a^3 * r^3 = -1
Expressing Conditions
1. The sum can be simplified:
a(1 + r + r^2) = 13/12
2. The product gives us:
a^3 * r^3 = -1
This implies that a = -1/(r^3).
Substituting the Product into the Sum
Substituting a into the sum equation gives:
(-1/(r^3))(1 + r + r^2) = 13/12
This can be rearranged to form a cubic equation in terms of r.
Solving for 'r'
Multiplying through by -r^3 yields:
-(1 + r + r^2) = (13/12)r^3
This leads to:
12r^3 + 12r^2 + 12r + 13 = 0.
Finding the Roots
Using numerical methods or the Rational Root Theorem, we can find the roots of this cubic equation.
Assuming r = -1 as a potential root, we can verify:
- Substitute r = -1 into the cubic equation to check if it balances.
- Once 'r' is found, substitute it back to find 'a'.
Final Terms and Common Difference
1. After solving, we will have values for 'a' and 'r'.
2. The terms are:
a, ar, ar^2.
3. The common difference of the G.P. is not defined as in arithmetic progression; however, the common ratio is 'r'.
This detailed approach provides a clear pathway to finding the terms of the G.P. and understanding their relationships through the given conditions.