Here are
geometric progression formulas.
p + q = n
p - q = m
We add the both equations and get
(p + q) + (p - q) = n + m
2p = n + m
p = (n + m)/2
we subtract the both simultaneous equations and we get
(p + q) - (p - q) = n - m
2q = n - m
q = (n - m)/2
As we see we never use the condition that both p and q form a geometric progression