10. (i) A sequence of positive numbers is defined by

$$\begin{aligned}
u _ { 1 } & = 5 \\
u _ { n + 1 } & = 3 u _ { n } + 2 , \quad n \geqslant 1
\end{aligned}$$

Prove by induction that, for $n \in \mathbb { Z } ^ { + }$,

$$u _ { n } = 2 \times ( 3 ) ^ { n } - 1$$

(ii) Prove by induction that, for $n \in \mathbb { Z } ^ { + }$,

$$\sum _ { r = 1 } ^ { n } \frac { 4 r } { 3 ^ { r } } = 3 - \frac { ( 3 + 2 n ) } { 3 ^ { n } }$$

\hfill \mbox{\textit{Edexcel F1 2016 Q10 [11]}}