Let S be a set of n distinct real numbers.

Let A

_{S} 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 A_{s}?

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

**k= 2n-3**

*Edited on ***March 18, 2016, 4:36 am**