Step-by-step Solution:

We are given that the terms 1, \( \log_y(x) \), \( \log_z(y) \), and \( -15 \log_x(z) \) 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: \[ a = 1, \quad b = \log_y(x), \quad c = \log_z(y), \quad d = -15 \log_x(z) \] From the AP property, we get: \[ \log_y(x) - 1 = \log_z(y) - \log_y(x) = -15 \log_x(z) - \log_z(y) \] From previous calculations, we found that the common difference is \( d = -2 \), so: \[ \log_y(x) - 1 = -2 \] \[ \log_y(x) = -1 \] This implies: \[ y^{-1} = x \] \[ x = \frac{1}{y} \] Step 2: Finding the Value of \( xy \) \[ xy = \left( \frac{1}{y} \right) y = 1 \] Final Answer: \[ \boxed{1} \]