- (i) Prove by induction that, for \(n \in \mathbb { Z } ^ { + }\),
$$\sum _ { r = 1 } ^ { n } r ^ { 2 } ( 2 r - 1 ) = \frac { 1 } { 6 } n ( n + 1 ) \left( 3 n ^ { 2 } + n - 1 \right)$$
(ii) Prove by induction that, for \(n \in \mathbb { Z } ^ { + }\),
$$\left( \begin{array} { c c }
7 & - 12
3 & - 5
\end{array} \right) ^ { n } = \left( \begin{array} { c c }
6 n + 1 & - 12 n
3 n & 1 - 6 n
\end{array} \right)$$