8. (i) Prove by induction that, for \(n \in \mathbb { Z } ^ { + }\) $$\sum _ { r = 1 } ^ { n } \frac { 2 r + 1 } { r ^ { 2 } ( r + 1 ) ^ { 2 } } = 1 - \frac { 1 } { ( n + 1 ) ^ { 2 } }$$ (ii) A sequence of positive rational numbers is defined by $$\begin{aligned} u _ { 1 } & = 3
u _ { n + 1 } & = \frac { 1 } { 3 } u _ { n } + \frac { 8 } { 9 } , \quad n \in \mathbb { Z } ^ { + } \end{aligned}$$ Prove by induction that, for \(n \in \mathbb { Z } ^ { + }\) $$u _ { n } = 5 \times \left( \frac { 1 } { 3 } \right) ^ { n } + \frac { 4 } { 3 }$$