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 :

Question

Accepted Answer

To solve this problem, we need to analyze the condition given in the question: each term in the geometric progression equals the sum of the next two terms. Let's denote the first term of the geometric progression as $ a $ and the common ratio as $ r $. The terms of the progression can be expressed as $ a, ar, ar^2, ar^3, \ldots $.

According to the problem, each term equals the sum of the next two terms. Therefore, for the first term $ a $, we have:

$$ a = ar + ar^2 $$

Dividing the entire equation by $ a $ (since $ a \neq 0 $), we get:

$$ 1 = r + r^2 $$

This is a quadratic equation in terms of $ r $:

$$ r^2 + r - 1 = 0 $$

To find the roots of this quadratic equation, we use the quadratic formula:

$$ r = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $$

Here, $ a = 1 $, $ b = 1 $, and $ c = -1 $. Plugging these values into the formula gives:

$$ r = \frac{-1 \pm \sqrt{1^2 - 4 \cdot 1 \cdot (-1)}}{2 \cdot 1} $$
$$ r = \frac{-1 \pm \sqrt{1 + 4}}{2} $$
$$ r = \frac{-1 \pm \sqrt{5}}{2} $$

The two possible values for $ r $ are:

$$ r = \frac{-1 + \sqrt{5}}{2} \quad \text{and} \quad r = \frac{-1 - \sqrt{5}}{2} $$

Since the progression consists of positive terms, we choose the positive root:

$$ r = \frac{-1 + \sqrt{5}}{2} $$

Now, we need to verify which of the given options matches this value. It is known from trigonometric identities that:

$$ 2 \sin 18^\circ = \frac{-1 + \sqrt{5}}{2} $$

Therefore, the common ratio $ r $ is indeed $ 2 \sin 18^\circ $, which corresponds to Option 4.

**Explanation for why other options are incorrect:**

1. **Option 1: $ \sin 18^\circ $** - This value does not match the derived common ratio $ \frac{-1 + \sqrt{5}}{2} $.

2. **Option 2: $ 2 \cos 18^\circ $** - This value does not match the derived common ratio $ \frac{-1 + \sqrt{5}}{2} $.

3. **Option 3: $ \cos 18^\circ $** - This value does not match the derived common ratio $ \frac{-1 + \sqrt{5}}{2} $.

Thus, the only option that correctly matches the common ratio derived from the given condition is Option 4: $ 2 \sin 18^\circ $.