Syzygies of adjoint linear series on projective varieties

Let X be a smooth complex projective variety of dimension n and let A be an ample and basepoint free divisor. We prove $K_X+mA$ satisfies property $N_p$ for $m⩾n+1+p$. We also show the graded ring of sections $R(X,K_X+mA)$ is Koszul for $m⩾n+2$.