Five positive integers A, B, C, D and E, with A < B < C < D < E, are such that:
(i) A, B and C (in this order) are in arithmetic sequence, and:
(ii) B, C and D (in his order) are in geometric sequence, and:
(iii) C, D and E (in this order) are in harmonic sequence.
Determine the minimum value of (E-A) such that there are precisely two quintuplets (A, B, C, D, E) that satisfy all the given conditions.
Note: Try to solve this problem analytically, although computer program/ spreadsheet solutions are welcome.
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