Let S be a set of n distinct real numbers.
be the set of numbers that occur as averages
of two distinct elements of S.
For a given n >= 2, what is the smallest possible number
of distinct elements in As?
Any arithmetic sequence a1, a2, ... can be mapped into sequence of first n integers 1,2, ...n. The set of the pairwise averages will be an arithmetic sequence A(s): 3/2, 2, 5/2, 3 ... n-1/2 i.e. all the numbers between the smallest and the biggest, including the ends spaced by 1/2.
If their number is k then there are k-1 spaces which should equal to (n-1/2-3/2)/(1/2)=2n-2
so k-1= 2n-2 and
Edited on March 18, 2016, 4:36 am