Why, if $X_1,...,X_n$ are pairwise independent, is $\sum_{i=1}^{k} X_i$ independent from $X_{k+1}$?

Question

Let $X_1,...,X_n$ be n pairwise independent random variables. Is $\sum_{i=1}^{k} X_i$ indep. from $X_{k+1}$ for some $1\leq k\lt n$?

[A proof I'm working through implicitly assumes that this is true]

Answer 1

Let $X_1, X_2$ be iid uniform on $\{-1,1\}$, let $X_3=X_1 X_2$. They are pairwise independent but $X_1+X_2$ is not independent of $X_3$. (If $X_1+X_2=0$, for instance, then $X_2=-X_1$ and $P(X_3=-1|X_1+X_2=0)=1,$ for instance.)