- (i) Prove by induction that, for \(n \in \mathbb { Z } ^ { + }\),
$$\left( \begin{array} { r r }
1 & 0
- 1 & 5
\end{array} \right) ^ { n } = \left( \begin{array} { c c }
1 & 0
- \frac { 1 } { 4 } \left( 5 ^ { n } - 1 \right) & 5 ^ { n }
\end{array} \right)$$
(ii) Prove by induction that, for \(n \in \mathbb { Z } ^ { + }\),
$$\sum _ { r = 1 } ^ { n } ( 2 r - 1 ) ^ { 2 } = \frac { 1 } { 3 } n \left( 4 n ^ { 2 } - 1 \right)$$