Conditional expectation of a summand given future partial sums

For independent and identically distributed random variables $X_j$ that are integrable random variables, with $S_n={\sum }_{j=1}^n{X}_j$, symmetry gives $\mathbb{E}[X_1\mid S_n,S_{n+1},\dots ]=S_n/n$. The decreasing sigma-algebras generated by these future sums permit application of the reverse martingale convergence theorem.