Geometric Progression

Since , , and are in geometric progression (GP), we have the relation: This is the first key equation. Next, we are given that the three differences of logarithms: are in arithmetic progression. For these differences to be in A.P., the middle term must be the average of the other two. Therefore: Simplifying the equation: Thus, we get: We also know that , so substituting into this: So, we have the relations:

Step 2: Triangle Inequality

For , , and to form the sides of a triangle, they must satisfy the triangle inequality: 1. 2. 3. Let's check these inequalities using and : 1. : This inequality is true. 2. : This inequality is true as well. 3. : This inequality is true.

Step 3: Type of Triangle

From the relations and , the sides of the triangle satisfy the triangle inequality. Now, check if the triangle is **right-angled**. For a right-angled triangle, the Pythagorean theorem should hold: Substitute and into this equation: This is clearly not true. Therefore, the triangle is **not right-angled**. Given that the Pythagorean theorem does not hold, but the triangle inequality does, the triangle formed by , , and is **obtuse-angled**.

Answer:

Step 1: Given positive real numbers such that:


Step 2: We want to minimize the expression:


Step 3: By the AM ≥ GM inequality, for any positive real numbers:


Step 4: Using the given condition:


So:


Step 5: Plug into the inequality:


Step 6: Multiply both sides by :


Step 7: The minimum value is therefore:


Step 8: If , then:
If , then this simplifies to .

Final Answer: (B):