Let
\(S_{N}=\sum_{r=1}^{N}(3 r+1)(3 r+4) \quad \text { and } \quad T_{N}=\sum_{r=1}^{N} \frac{1}{(3 r+1)(3 r+4)} .\)
(i) Use standard results from the List of Formulae (MF10) to show that
\(S_{N}=N\left(3 N^{2}+12 N+13\right)\)

(ii) Use the method of differences to show that
\(T_{N}=\frac{1}{12}-\frac{1}{3(3 N+4)}\)
(iii) Deduce that \(\frac{S_{N}}{T_{N}}\) is an integer.

(iv) Find \(\lim _{N \rightarrow \infty} \frac{S_{N}}{N^{3} T_{N}}\).