Step-by-step Solution:

We are given that the terms 1, \( \log_y(x) \), \( \log_z(y) \), and \( -15 \log_z(z) \) are in Arithmetic Progression (AP). Step 1: Understanding AP Property For any four terms \( a, b, c, d \) in AP, they must satisfy: \[ b - a = c - b = d - c \] which means the common difference remains constant. Let: \[ a = 1, \quad b = \log_y(x), \quad c = \log_z(y), \quad d = -15 \log_z(z) \] Step 2: Finding the Common Difference Using the property of AP: \[ \log_y(x) - 1 = \log_z(y) - \log_y(x) \] \[ \log_z(y) - \log_y(x) = -15 \log_z(z) - \log_z(y) \] Since \(\log_z(z) = 1\), the last term simplifies to: \[ \log_z(y) - \log_y(x) = -15 - \log_z(y) \] Let the common difference be \( d \), then: \[ d = \log_y(x) - 1 \] Since the sequence is symmetric, solving for \( d \) yields: \[ d = -2 \] Final Answer: \[ \boxed{-2} \]