Five positive integers A
, with A
, are such that:
(in this order) are in harmonic sequence
(in this order) are in geometric sequence, and:
(in this order) are in arithmetic sequence.
Determine the minimum value of (E
) such that there are precisely three quintuplets (A
) that satisfy all the given conditions.
I agree with Jer, as far as he goes:
C = (AB)/(2A-B)
D = (A^2B)/(2A-B)^2
E = (AB^2)/(2A-B)^2
Let b be the difference constant in the harmonic series; since we know C, A and B can then be represented as
The smallest quintuplets I have found having a common difference (E-A) are:
A b B C c D d E E-A
6 5 11 66 6 396 330 726 720
30 20 50 150 3 450 300 750 720
90 45 135 270 2 540 270 810 720
(b,c,d, are the difference constants) but I lack the facilities for an exhaustive search, and this result involves some 'lateral thinking' about the sort of results that might work.
Posted by broll
on 2011-04-05 12:17:17