3 The sequence of positive numbers $u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots$ is such that $u _ { 1 } < 3$ and, for $n \geqslant 1$,

$$u _ { n + 1 } = \frac { 4 u _ { n } + 9 } { u _ { n } + 4 }$$

(i) By considering $3 - u _ { n + 1 }$, or otherwise, prove by mathematical induction that $u _ { n } < 3$ for all positive integers $n$.\\

(ii) Show that $u _ { n + 1 } > u _ { n }$ for $n \geqslant 1$.\\

\hfill \mbox{\textit{CAIE FP1 2018 Q3}}