We define the subweighted value $x$ of an ordered set $A$ with $k$ elements if for each element $a_i$ in $A$ we alternate the sum of its indexed weight; that is $x = \sum_{i=1}^{k} ((-1)^{i+1} \cdot i \cdot a_i)$. Find the expected value of the subweighted value of a randomly chosen subset of a ordered set $S$ with $n$ elements (any subset is equally likely to be chosen).
I ran a short code and it only seems like this value would depend on the first two values; it is solely determined by $s_1$ and $s_2$, the first two values when $S$ is ordered from least to greatest. Is there a way to show this rigorously?
For example when the set is $\{1,2\}$ or $\{1,2,3\}$ or $\{1,2,923,203\}$, the expected value is still $0$.