  Let $G$ be a finite abelian group and let $A\subseteq \mathbb{Z}$ be
nonempty. Let $D_A(G)$ denote the minimal integer such that any sequence over
$G$ of length $D_A(G)$ must contain a nontrivial subsequence $s_1... s_r$ such
that $\sum_{i=1}^{r}w_is_i=0$ for some $w_i\in A$. Let $E_A(G)$ denote the
minimal integer such that any sequence over $G$ of length $E_A(G)$ must contain
a subsequence of length $|G|$, $s_1... s_{|G|}$, such that
$\sum_{i=1}^{|G|}w_is_i=0$ for some $w_i\in A$. In this paper, we show that
$$E_A(G)=|G|+D_A(G)-1,$$ confirming a conjecture of Thangadurai and the
expectations of Adhikari, et al. The case $A=\{1\}$ is an older result of Gao,
and our result extends much partial work done by Adhikari, Rath, Chen, David,
Urroz, Xia, Yuan, Zeng and Thangadurai. Moreover, under a suitable multiplicity
restriction, we show that not only can zero be represented in this manner, but
an entire nontrivial subgroup, and if this subgroup is not the full group $G$,
we obtain structural information for the sequence generalizing another
non-weighted result of Gao. Our full theorem is valid for more general $n$-sums
with $n\geq |G|$, in addition to the case $n=|G|$.
