The numbers 2^x, 2^y and 2^z are the terms of a geometric progression. We have to determine x, y and z which are the terms of an arithmetic progression.
As the consecutive terms of a geometric progression have a common ratio : 2^y/2^x = 2^z/2^y
=> 2^(y - x) = 2^(z - y)
=> y - x = z - y
=> 2y = x + z
This is true for all values of x, y and z that are consecutive terms of an arithmetic progression.
x, y and z do not have a unique value, they can be any three consecutive terms of any arithmetic progression.
If x,y,z are the terms of an AP, we can apply the mean value theorem:
y = (x+z)/2
We'll cross multiply and we'll get:
2y = (x+z) (1)
If 2^x,2^y,2^z are the terms of a GP then 2^y is the geometric mean of 2^x and 2^z:
2^y = sqrt(2^x*2^z)
We'll square raise both sides:
2^2y = 2^x*2^z
Since the bases of the exponentials from the right side are matching, we'll add the exponents:
2^2y = 2^(x+z)
Since the bases are matching, we'll apply one to one property:
2y = (x+z)
Therefore, the integer terms x,y,and z may have any value, as long as they respect the constraint 2y = (x+z).