In a G.P. of real numbers, the sum of the first two terms is 7. The sum of the first six terms is 91. The sum of the first four terms is`"______________"`

Answer

To solve the problem step by step, we will use the properties of a geometric progression (G.P.).

### Given:
1. The sum of the first two terms is 7.
2. The sum of the first six terms is 91.

### Let:
- The first term of the G.P. be \( A \).
- The common ratio be \( R \).

### Step 1: Write the equations based on the given information.
The sum of the first two terms can be expressed as:
\[ A + AR = 7 \quad \text{(1)} \]

The sum of the first six terms can be expressed using the formula for the sum of a G.P.:
\[ S_n = A \frac{1 - R^n}{1 - R} \quad \text{for } R \neq 1 \]
Thus, the sum of the first six terms is:
\[ S_6 = A \frac{1 - R^6}{1 - R} = 91 \quad \text{(2)} \]

### Step 2: Simplify the equations.
From equation (1), we can factor out \( A \):
\[ A(1 + R) = 7 \quad \Rightarrow \quad A = \frac{7}{1 + R} \quad \text{(3)} \]

### Step 3: Substitute \( A \) from equation (3) into equation (2).
Substituting \( A \) into equation (2):
\[ \frac{7}{1 + R} \cdot \frac{1 - R^6}{1 - R} = 91 \]

### Step 4: Cross-multiply to eliminate the fraction.
\[ 7(1 - R^6) = 91(1 - R)(1 + R) \]
Expanding both sides:
\[ 7 - 7R^6 = 91(1 - R^2) \]
\[ 7 - 7R^6 = 91 - 91R^2 \]

### Step 5: Rearranging the equation.
Rearranging gives:
\[ 7R^6 - 91R^2 + 84 = 0 \]

### Step 6: Let \( x = R^2 \) to simplify the equation.
This transforms our equation into:
\[ 7x^3 - 91x + 84 = 0 \]

### Step 7: Solve for \( x \).
Using the Rational Root Theorem or synthetic division, we can find the roots of this cubic equation. After testing possible rational roots, we find that \( x = 3 \) is a root.

### Step 8: Factor the cubic equation.
Using synthetic division, we can factor the cubic polynomial:
\[ 7x^3 - 91x + 84 = (x - 3)(7x^2 + 21x - 28) \]

### Step 9: Solve the quadratic equation.
Using the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{-21 \pm \sqrt{21^2 - 4 \cdot 7 \cdot (-28)}}{2 \cdot 7} \]
Calculating the discriminant:
\[ 21^2 + 784 = 441 + 784 = 1225 \]
Thus:
\[ x = \frac{-21 \pm 35}{14} \]
Calculating the two possible values for \( x \):
1. \( x = 1 \) (valid)
2. \( x = -4 \) (not valid since \( x = R^2 \) must be non-negative)

### Step 10: Find \( R \) and \( A \).
Since \( x = R^2 = 3 \), we have \( R = \sqrt{3} \). Now substituting back to find \( A \):
\[ A(1 + \sqrt{3}) = 7 \quad \Rightarrow \quad A = \frac{7}{1 + \sqrt{3}} \]

### Step 11: Calculate the sum of the first four terms.
The sum of the first four terms is given by:
\[ S_4 = A(1 + R + R^2 + R^3) \]
We can express \( S_4 \) as:
\[ S_4 = A \cdot \frac{1 - R^4}{1 - R} \]
Substituting \( A \) and simplifying gives:
\[ S_4 = \frac{7}{1 + \sqrt{3}} \cdot \frac{1 - (\sqrt{3})^4}{1 - \sqrt{3}} = \frac{7}{1 + \sqrt{3}} \cdot \frac{1 - 9}{1 - \sqrt{3}} = \frac{7}{1 + \sqrt{3}} \cdot \frac{-8}{1 - \sqrt{3}} \]
Calculating this gives:
\[ S_4 = 28 \]

### Final Answer:
The sum of the first four terms is \( \boxed{28} \).