- (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 \(n \in \mathbb { Z } ^ { + }\),
$$\sum _ { r = 1 } ^ { n } \frac { 4 r } { 3 ^ { r } } = 3 - \frac { ( 3 + 2 n ) } { 3 ^ { n } }$$
\(\_\_\_\_\) VJYV SIHI NI JIIYM ION OO \(\quad\) VJYV SIHI NI JIIYM ION OC \(\quad\) VJYV SIHI NI JLIVM ION OC
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