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.

One sequence (no proof of minimum value) that follows the requirements given by (i), (ii) and (iii) is: (2, 4, 6, 9, 18).

(i) The common difference between A, B and C is +2.
(ii) The common ratio between B, C and D is +3/2.
(iii) the common difference between the reciprocals of C, D and E is -1/18.

Edited on October 28, 2008, 10:43 am

Posted by Dej Mar on 2008-10-28 10:38:15