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.
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
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Posted by Charlie
on 2008-10-28 18:58:53 |
computer solutions -- no proof
