Each of A, B, C and D is a positive integer with A < B < C < D
having gcd(A, B, C, D) = 1 such that:

(i) A, B and C are in geometric sequence, and:
(ii) B, C and D are in arithmetic sequence, and:
(iii) A, B and D are in harmonic sequence.

Does there exist an infinite number of quadruplets satisfying the given conditions? Give reasons for your answer.