9. (i) A sequence of numbers \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is defined by $$u _ { n + 1 } = \frac { 1 } { 3 } \left( 2 u _ { n } - 1 \right) \quad u _ { 1 } = 1$$ Prove by induction that, for \(n \in \mathbb { Z } ^ { + }\) $$u _ { n } = 3 \left( \frac { 2 } { 3 } \right) ^ { n } - 1$$ (ii) \(\mathrm { f } ( n ) = 2 ^ { n + 2 } + 3 ^ { 2 n + 1 }\) Prove by induction that, for \(n \in \mathbb { Z } ^ { + } , \mathrm { f } ( n )\) is a multiple of 7