If in a geometric progression consisting of positive terms, each term equals the sum of the next two terms, then the common ratio of this progression equals

Options

A. `sqrt(5)`
B. `(1)/(2) (sqrt(5)-1)`
C. `(1)/(2) (1- sqrt(5))`
D. `(1)/(2) sqrt(5)`

Answer

To solve the problem, we need to find the common ratio \( r \) of a geometric progression (GP) where each term equals the sum of the next two terms.

### Step-by-Step Solution:

1. **Define the terms of the GP**:
Let the first term of the GP be \( a \). Then the terms can be expressed as:
- First term: \( a \)
- Second term: \( ar \)
- Third term: \( ar^2 \)
- Fourth term: \( ar^3 \)
- And so on...

2. **Set up the equation based on the given condition**:
According to the problem, each term equals the sum of the next two terms. Therefore, we can write:
\[ a = ar + ar^2 \]

3. **Simplify the equation**:
We can factor out \( a \) from the right side:
\[ a = a(r + r^2) \]
Since \( a \) is positive and not zero, we can divide both sides by \( a \):
\[ 1 = r + r^2 \]

4. **Rearrange the equation**:
Rearranging gives us a standard quadratic equation:
\[ r^2 + r - 1 = 0 \]

5. **Apply the quadratic formula**:
The quadratic formula is given by:
\[ r = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 1 \), \( b = 1 \), and \( c = -1 \). Plugging in these values:
\[ r = \frac{-1 \pm \sqrt{1^2 - 4 \cdot 1 \cdot (-1)}}{2 \cdot 1} \]
Simplifying further:
\[ r = \frac{-1 \pm \sqrt{1 + 4}}{2} = \frac{-1 \pm \sqrt{5}}{2} \]

6. **Determine the positive solution**:
Since the terms of the GP are positive, we take the positive root:
\[ r = \frac{-1 + \sqrt{5}}{2} \]

7. **Final answer**:
Thus, the common ratio of the progression is:
\[ r = \frac{\sqrt{5} - 1}{2} \]