**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.