Let $n$ be a positive integer greater than or equal to $6$, and suppose that $a_1, a_2, ...,a_n$ are real numbers such that the sums $a_i + a_j$ for $1 \le i<j\le n$, taken in some order, form consecutive terms of an arithmetic progression $A$, $A + d$, $...$ ,$A + (k-1)d$, where $k = n(n-1)/2$. What are the possible values of $d$?