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 Geometric Progression (GP) Properties.
If \( a, b, c \) form a geometric progression, then:\[\frac{b}{a} = \frac{c}{b}\]which is equivalent to:\[b^2 = ac\]

Step 2: Apply the Given Condition.
The condition provided is \( a^x = b^y = c^z \). This means:\[a^x = b^y = c^z = k (\text{for some constant } k)\]Taking the logarithm of each equation yields:\[x \log a = y \log b = z \log c\]

Step 3: Derive the Relationship.
From GP properties, \( \log b = \frac{1}{2} (\log a + \log c) \). This implies that \( x, y, z \) form an arithmetic progression. Therefore, the correct classification is 1. Arithmetic Progression.