3 (a) By considering $( 2 r + 1 ) ^ { 3 } - ( 2 r - 1 ) ^ { 3 }$, use the method of differences to prove that

$$\sum _ { r = 1 } ^ { n } r ^ { 2 } = \frac { 1 } { 6 } n ( n + 1 ) ( 2 n + 1 )$$

Let $S _ { n } = 1 ^ { 2 } + 3 \times 2 ^ { 2 } + 3 ^ { 2 } + 3 \times 4 ^ { 2 } + 5 ^ { 2 } + 3 \times 6 ^ { 2 } + \ldots + \left( 2 + ( - 1 ) ^ { n } \right) n ^ { 2 }$.\\
(b) Show that $\mathrm { S } _ { 2 \mathrm { n } } = \frac { 1 } { 3 } \mathrm { n } ( 2 \mathrm { n } + 1 ) ( \mathrm { an } + \mathrm { b } )$, where $a$ and $b$ are integers to be determined.\\

(c) State the value of $\lim _ { n \rightarrow \infty } \frac { S _ { 2 n } } { n ^ { 3 } }$.\\

\hfill \mbox{\textit{CAIE Further Paper 1 2022 Q3}}