Let the sum of first three terms of G.P. with real terms be 13/12 and their product is -1. If the absolute value of the sum of their infinite terms is `S ,` then the value of `7S` is ______.

Explanation

To solve the problem step by step, we will denote the three terms of the G.P. as \( A, Ar, Ar^2 \).

### Step 1: Set up the equations
We know from the problem statement:
1. The sum of the first three terms is given by:
\[ A + Ar + Ar^2 = 13/12 \]
2. The product of the three terms is given by:
\[ A \cdot Ar \cdot Ar^2 = A^3 r^3 = -1 \]

### Step 2: Solve for \( A \)
From the product equation, we can express \( A \):
\[ A^3 = -1 \implies A = -1 \]

### Step 3: Substitute \( A \) into the sum equation
Substituting \( A = -1 \) into the sum equation:
\[ -1 + (-1)r + (-1)r^2 = 13/12 \]
This simplifies to:
\[ -1 - r - r^2 = 13/12 \]
Rearranging gives:
\[ -r - r^2 = 13/12 + 1 = 13/12 + 12/12 = 25/12 \]
Thus:
\[ r^2 + r + \frac{25}{12} = 0 \]

### Step 4: Clear the fraction
Multiply through by 12 to eliminate the fraction:
\[ 12r^2 + 12r + 25 = 0 \]

### Step 5: Solve the quadratic equation
Now we can apply the quadratic formula \( r = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
Here, \( a = 12, b = 12, c = 25 \):
\[ r = \frac{-12 \pm \sqrt{12^2 - 4 \cdot 12 \cdot 25}}{2 \cdot 12} \]
Calculating the discriminant:
\[ 12^2 - 4 \cdot 12 \cdot 25 = 144 - 1200 = -1056 \]
Since the discriminant is negative, we have complex roots. However, we are interested in real terms, so we will consider the roots:
\[ r = \frac{-12 \pm i\sqrt{1056}}{24} \]
This gives us two complex values for \( r \).

### Step 6: Find the sum of infinite terms
The formula for the sum of an infinite G.P. is:
\[ S = \frac{A}{1 - r} \]
Substituting \( A = -1 \) and \( r = -\frac{3}{4} \) (the valid real root):
\[ S = \frac{-1}{1 - (-\frac{3}{4})} = \frac{-1}{1 + \frac{3}{4}} = \frac{-1}{\frac{7}{4}} = -\frac{4}{7} \]

### Step 7: Calculate \( 7S \)
Now we calculate \( 7S \):
\[ 7S = 7 \times -\frac{4}{7} = -4 \]

### Final Answer
Thus, the value of \( 7S \) is:
\[ \boxed{-4} \]