In this paper we establish a central limit theorem for weighted sums of $Y_n={sum_{i=1}^{n}}a_n,_iX_i$, where ${a_{n,i},;n{in}N,;1{leq}i{leq}n}$ is an array of nonnegative numbers such that ${sup}_{n{geq}1}{sum_{i=1}^{n}}a_{n,i}^2$ < ${infty}$, ${max}_{1{leq}i{leq}n}a_{n,i}{ ightarrow}0$ and ${X_i,;i{in}N}$ is a sequence of linear negatively quadrant dependent random variables with $EX_i=0$ and $EX_i^2$ < ${infty}$. Using this result we will obtain a central limit theorem for partial sums of linear processes.