A set of six positive integers contains an arithmetic sequence of four terms, a geometric sequence of four terms, and a harmonic sequence of four terms. What are the numbers in the set when the largest member of the set is a minimum?
Note: A harmonic sequence is a sequence whose reciprocals form an arithmetic sequence. ex: ({10, 12, 15, 20} is harmonic since {1/10, 1/12, 1/15, 1/20} is arithmetic)
There are precisely three possible sets (S) of six positive whole
numbers which satisfy conditions of the problem and these are:
(I) S = {3, 4, 5, 6, 12, 24} ; arirhmetic sequence (A) = {3, 6, 12, 24}; geometric sequence (G) = {3, 6, 12, 24}; harmonic sequence(H) = {3, 4, 6, 12}
(II) S = {3, 4, 6, 9, 12, 24} ; A = {3, 6, 9, 12};
G = {3, 6, 12, 24}; H= {3, 4, 6, 12)
(III) S = {3, 4, 6, 12, 18, 24} ; A = {6, 12, 18, 24};
G = {3, 6, 12, 24}; H= {3, 4, 6, 12);
Edited on March 18, 2008, 11:11 am
Answer
