If every term of a G.P with positive terms is the sum of its two previous terms, then the common ratio of the G.P is

Options

A. 1−5–√2
B. √+12
C. 9
D. None of these

Answer

To solve the problem, we need to find the common ratio \( R \) of a geometric progression (G.P.) where each term is the sum of the two previous terms. Let's break down the solution step by step.

### Step 1: Define the Terms of the G.P.
Let the first term of the G.P. be \( A \). The terms of the G.P. can be expressed as:
- First term: \( A \)
- Second term: \( AR \)
- Third term: \( AR^2 \)
- Fourth term: \( AR^3 \)
- And so on...

### Step 2: Set Up the Equation Based on the Given Condition
According to the problem, every term is the sum of its two previous terms. We can start with the third term:
\[ AR^2 = A + AR \]

### Step 3: Simplify the Equation
We can factor out \( A \) from the right side:
\[ AR^2 = A(1 + R) \]
Assuming \( A \neq 0 \) (since all terms are positive), we can divide both sides by \( A \):
\[ R^2 = 1 + R \]

### Step 4: Rearrange the Equation
Rearranging the equation gives us a standard quadratic form:
\[ R^2 - R - 1 = 0 \]

### Step 5: Solve the Quadratic Equation
We can apply the quadratic formula \( R = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) where \( a = 1, b = -1, c = -1 \):
\[ R = \frac{-(-1) \pm \sqrt{(-1)^2 - 4 \cdot 1 \cdot (-1)}}{2 \cdot 1} \]
\[ R = \frac{1 \pm \sqrt{1 + 4}}{2} \]
\[ R = \frac{1 \pm \sqrt{5}}{2} \]

### Step 6: Determine Valid Solutions
This gives us two potential solutions for \( R \):
1. \( R = \frac{1 + \sqrt{5}}{2} \)
2. \( R = \frac{1 - \sqrt{5}}{2} \)

Since \( \sqrt{5} \approx 2.236 \), the second solution \( \frac{1 - \sqrt{5}}{2} \) will be negative, which is not acceptable since all terms of the G.P. are positive. Therefore, we accept only:
\[ R = \frac{1 + \sqrt{5}}{2} \]

### Step 7: Conclusion
Now we check the options provided in the question. Since \( \frac{1 + \sqrt{5}}{2} \) does not match any of the given options, we conclude that the correct answer is:
**None of these (Option D)**.