$\text{ The largest terms of the sequence }\frac{1}{503},\frac{4}{524},\frac{9}{581},\frac{16}{692},\dots \text{ is }$

${\mathrm{T}}_{\mathrm{n}}=\frac{{\mathrm{n}}^2}{500+3{\mathrm{n}}^3}=\mathrm{f}\left(\mathrm{n}\right)$

${\mathrm{f}}^1\left(\mathrm{n}\right)=\frac{\mathrm{n}\left(1000-3{\mathrm{n}}^3\right)}{{\left(500+3{\mathrm{n}}^3\right)}^2}$

$\begin{array}{l}\text{ For the maxima or }min{\mathrm{f}}^1\left(\mathrm{n}\right)=0\\ \mathrm{n}={\left(\frac{1000}{3}\right)}^{1/3}\\ \text{ Now }6<{\left(\frac{1000}{3}\right)}^{1/3}<7\end{array}$

$\therefore {\mathrm{T}}_7\text{ is the largest term. }$

$\text{ the largest term of the sequence }=\frac{49}{500+1029}=\frac{49}{1529}$