In a GP of positive terms, any term is equal to one-third of the sum of next two terms. What is the common ratio of the GP?

Options

A. `(sqrt(13)+1)/(2)`
B. `(sqrt(13)-1)/(2)`
C. `(sqrt(13)+1)/(3)`
D. `sqrt(13)`

Explanation

To solve the problem, we need to find the common ratio of a geometric progression (GP) where any term is equal to one-third of the sum of the next two terms. Let's denote the first term of the GP as \( A \) and the common ratio as \( R \). The terms of the GP can be expressed as \( A, AR, AR^2, \ldots \).

### Step-by-Step Solution:

1. **Identify the terms**:
Let's denote the first term as \( A \), the second term as \( AR \), and the third term as \( AR^2 \).

2. **Set up the equation**:
According to the problem, any term (let's take the first term \( A \)) is equal to one-third of the sum of the next two terms:
\[ A = \frac{1}{3}(AR + AR^2) \]

3. **Simplify the equation**:
We can simplify the right-hand side:
\[ A = \frac{1}{3}A(R + R^2) \]

4. **Eliminate \( A \)**:
Since \( A \) is a positive term, we can safely divide both sides by \( A \) (as \( A \neq 0 \)):
\[ 1 = \frac{1}{3}(R + R^2) \]

5. **Multiply through by 3**:
To eliminate the fraction, multiply both sides by 3:
\[ 3 = R + R^2 \]

6. **Rearrange the equation**:
Rearranging gives us a standard quadratic equation:
\[ R^2 + R - 3 = 0 \]

7. **Use the quadratic formula**:
The quadratic formula is given by:
\[ R = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 1, b = 1, c = -3 \).

8. **Calculate the discriminant**:
First, calculate the discriminant \( D \):
\[ D = b^2 - 4ac = 1^2 - 4 \cdot 1 \cdot (-3) = 1 + 12 = 13 \]

9. **Find the values of \( R \)**:
Substitute the values into the quadratic formula:
\[ R = \frac{-1 \pm \sqrt{13}}{2} \]

10. **Select the positive root**:
Since we are looking for positive terms in the GP, we only consider the positive root:
\[ R = \frac{-1 + \sqrt{13}}{2} \]

### Final Answer:
The common ratio \( R \) of the GP is:
\[ R = \frac{-1 + \sqrt{13}}{2} \]