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?
S consists of real numbers, so its members can be ordered. Let k_1, k_2, k_3, ... k_n be the members of S sorted in ascending order. k_1 is the smallest member and k_n is the largest member.
Let A_k1 be the subset of A_s consisting of all n-1 averages using k_1. Similarly let A_kn be the subset of A_s consisting of all n-1 averages using k_n. These two subsets have one member in common: (k_1+k_n)/2. All the other members of A_k1 are smaller than this value and all the other members of A_kn are larger than this value.
The union of A_k1 and A_kn has (n-1)+(n-1)-1 = 2n-3 members. The union also must be a subset of A_s. Therefore A_s must have at least 2n-3 members.
As Charlie noted in his solution, if S is an arithmetic sequence of n terms then A_s has 2n-3 members. This shows the theoretical minimum size of A_s, as determined above, can be achieved for some set S. The smallest possible number of distinct members if A_s is 2n-3.