Let {$X_{nk}$mid$1;{leq};k;{leq};n,;n;{geq};1$</TEX>} be an array of row negatively associated (NA) random variables which satisfy $P($mid$X_{nk}$mid$;>;x);{leq};P($mid$X$mid$;>;x)$</TEX>. For weighed sums ${{Sigma}_{k=1}}^{Tn};a_kX_{nk}$ indexed by random variables {$T_n$mid$n;{geq}$1$}, we establish a general weak law of large numbers (WLLN) of the form $({{Sigma}_{k=1}}^{Tn};a_kX_{nk};-;v_{[nk]});/b_{[an]}$ under some suitable conditions, where ${a_n$mid$n;geq;1},; {b_n$mid$n;geq;1}$</TEX> are sequences of constants with $a_n;>;0,;0;<;b_n; ightarrow ;infty,;n;{geq};1$, and {$v_{an}$mid$n;{geq};1$</TEX>} is an array of random variables, and the symbol [x] denotes the greatest integer in x.