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Sequence Group III (Posted on 2008-10-28) Difficulty: 3 of 5
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.

See The Solution Submitted by K Sengupta    
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Solution computer solutions -- no proof | Comment 2 of 5 |

The only quintuplets meeting the criteria where A <= 200,000 and E - A < 100 are:

     A     B     C     D     E   E - A
     2     4     6     9    18    16
     6     9    12    16    24    18
    20    25    30    36    45    25
     4     8    12    18    36    32
    12    18    24    32    48    36
     6    12    18    27    54    48
    40    50    60    72    90    50
    18    27    36    48    72    54
     8    16    24    36    72    64
    36    48    60    75   100    64
    24    36    48    64    96    72
     3     9    15    25    75    72
    60    75    90   108   135    75
    10    20    30    45    90    80
   144   162   180   200   225    81
    30    45    60    80   120    90
    12    24    36    54   108    96
   126   147   168   192   224    98

So presumably the answer is E-A = 64, unless beyond A = 200,000 there's a third set where E-A = 64 or a second set for one of the lower E-A values. But nothing was found for any A higher than 144 (or actually, beyond 40 for what we're considering), so it's probably safe.

The results of the below program were sorted on E-A for the above table.

DEFDBL A-Z
OPEN "seqgrp3.txt" FOR OUTPUT AS #2
FOR a = 1 TO 200000
FOR b = a + 1 TO a + 100
  c = 2 * b - a
  d = c * c / b
  IF d = INT(d) AND c / b <> 2 THEN
    e = d * c / (2 * c - d)
    IF e = INT(e) AND e - a < 100 THEN
      PRINT #2, USING "######"; a; b; c; d; e; e - a
    END IF
  END IF
NEXT
NEXT
CLOSE

Edited on October 28, 2008, 6:59 pm
  Posted by Charlie on 2008-10-28 18:58:53

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