3 The sequence of positive numbers $u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots$ is such that $u _ { 1 } > 4$ and, for $n \geqslant 1$,

$$u _ { n + 1 } = \frac { u _ { n } ^ { 2 } + u _ { n } + 12 } { 2 u _ { n } }$$

(a) By considering $\mathrm { u } _ { \mathrm { n } + 1 } - 4$, or otherwise, prove by mathematical induction that $\mathrm { u } _ { \mathrm { n } } > 4$ for all positive integers $n$.\\

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

\hfill \mbox{\textit{CAIE Further Paper 1 2022 Q3}}