Question

If \( a, b, c \) are in Geometric Progression and \( a^x = b^y = c^z \), then \( x, y, z \) are in:

• A. Arithmetic Progression
• B. Geometric Progression

Hint:

When dealing with geometric progressions, use the logarithmic form to relate exponents and check if they form an arithmetic progression.

Answer 1

The Correct Option is A

Step 1: Recall the properties of a Geometric Progression (GP).
If \( a, b, c \) are in geometric progression, we know that: \[ \frac{b}{a} = \frac{c}{b} \] or equivalently, \[ b^2 = ac \]

Step 2: Use the given relationship.
The given condition is \( a^x = b^y = c^z \). This implies the following: \[ a^x = b^y = c^z = k (\text{for some constant } k) \] Taking logarithms of each equation: \[ x \log a = y \log b = z \log c \]

Step 3: Express the relationship.
From the properties of geometric progression, we know that \( \log b = \frac{1}{2} (\log a + \log c) \), so the relationships between \( x, y, z \) form an arithmetic progression. Therefore, the correct answer is 1. Arithmetic Progression.