The number of terms of the series $54,51,48,\mathrm{.......}$  such that their sum is $513$  is

$a=54,d=51-54=-3$

$S_n=513$

$\frac{n}{2}\left[2a+\left(n-1\right)d\right]=513$

$n\left[108-3n+3\right]=1026$

$n\left(111-3n\right)=1026$

$111n-3n^2=1026$

$3n^2-111n+1026=0$

$n^2-37n+342=0$

$n^2-18n-19n+342=0$

$\left(n-18\right)\left(n-19\right)=0$

$n=18$    $\left(or\right)$    $n=19$