Let `a_(1), a_(2), a_(3)` be three positive numbers which are in geometric progression with common ratio r. The inequality `a_(3) gt a_(2)+2a_(1)` holds true if r is equal to

Options

A. 2
B. 1.5
C. 0.5
D. 2.5

Answer

To solve the problem, we need to determine the value of the common ratio \( r \) for three positive numbers \( a_1, a_2, a_3 \) that are in geometric progression (GP) such that the inequality \( a_3 > a_2 + 2a_1 \) holds true.

### Step-by-Step Solution:

1. **Define the terms in GP**:
Since \( a_1, a_2, a_3 \) are in geometric progression with common ratio \( r \):
- Let \( a_1 = a \)
- Then \( a_2 = ar \)
- And \( a_3 = ar^2 \)

2. **Set up the inequality**:
We need to analyze the inequality:
\[ a_3 > a_2 + 2a_1 \]
Substituting the expressions for \( a_1, a_2, a_3 \):
\[ ar^2 > ar + 2a \]

3. **Simplify the inequality**:
Divide both sides by \( a \) (since \( a > 0 \)):
\[ r^2 > r + 2 \]

4. **Rearrange the inequality**:
Rearranging gives:
\[ r^2 - r - 2 > 0 \]

5. **Factor the quadratic**:
We can factor the left-hand side:
\[ (r - 2)(r + 1) > 0 \]

6. **Determine the critical points**:
The critical points from the factors are \( r = 2 \) and \( r = -1 \).

7. **Test intervals**:
We need to test the intervals determined by the critical points:
- Interval 1: \( r < -1 \)
- Interval 2: \( -1 < r < 2 \)
- Interval 3: \( r > 2 \)

**Testing the intervals**:
- For \( r < -1 \) (e.g., \( r = -2 \)):
\[ (-2 - 2)(-2 + 1) = (-4)(-1) = 4 > 0 \quad \text{(True)} \]
- For \( -1 < r < 2 \) (e.g., \( r = 0 \)):
\[ (0 - 2)(0 + 1) = (-2)(1) = -2 < 0 \quad \text{(False)} \]
- For \( r > 2 \) (e.g., \( r = 3 \)):
\[ (3 - 2)(3 + 1) = (1)(4) = 4 > 0 \quad \text{(True)} \]

8. **Conclusion**:
The inequality \( (r - 2)(r + 1) > 0 \) holds true for:
- \( r < -1 \) or \( r > 2 \)

Since \( r \) must be a positive number, we conclude that:
\[ r > 2 \]

### Testing the Options:
Given the options \( 2, 1.5, 0.5, 2.5 \):
- **Option 1: \( r = 2 \)**:
\[ (2 - 2)(2 + 1) = 0 \quad \text{(False)} \]
- **Option 2: \( r = 1.5 \)**:
\[ (1.5 - 2)(1.5 + 1) = (-0.5)(2.5) < 0 \quad \text{(False)} \]
- **Option 3: \( r = 0.5 \)**:
\[ (0.5 - 2)(0.5 + 1) = (-1.5)(1.5) < 0 \quad \text{(False)} \]
- **Option 4: \( r = 2.5 \)**:
\[ (2.5 - 2)(2.5 + 1) = (0.5)(3.5) > 0 \quad \text{(True)} \]

Thus, the correct answer is \( r = 2.5 \).