9. (a) Prove by induction that, for \(n \in \mathbb { Z } ^ { + }\), $$\sum _ { r = 1 } ^ { n } ( r + 1 ) 2 ^ { r - 1 } = n 2 ^ { n }$$ (b) A sequence of numbers is defined by $$\begin{gathered} u _ { 1 } = 0 , \quad u _ { 2 } = 32 ,
u _ { n + 2 } = 6 u _ { n + 1 } - 8 u _ { n } \quad n \geqslant 1 \end{gathered}$$ Prove by induction that, for \(n \in \mathbb { Z } ^ { + }\), $$u _ { n } = 4 ^ { n + 1 } - 2 ^ { n + 3 }$$