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).
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