Let 3, x, y, z be in Arithmetic Progression and 3, x – 1, y + 1, z + 9 be in Geometric Progression. Then, the arithmetic mean of x, y, and z is:

In an Arithmetic Progression (AP), the difference between any two successive terms is constant. Let's denote this common difference as 'd'. So, we can write the following equations based on the given AP:

x = 3 + d y = x + d = 3 + 2d z = y + d = 3 + 3d

In a Geometric Progression (GP), the ratio of any two successive terms is constant. Let's denote this common ratio as 'r'. So, we can write the following equations based on the given GP:

x - 1 = 3r y + 1 = (x - 1)r = 3r^2 z + 9 = (y + 1)r = 3r^3

Now, we have a system of 6 equations. We can solve this system to find the values of x, y, z, and then calculate their arithmetic mean.

However, this system is quite complex and may not have a simple solution. It may require the use of numerical methods or software to solve.