3 The sequence \(a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots\) is such that \(a _ { 1 } > 5\) and \(a _ { n + 1 } = \frac { 4 a _ { n } } { 5 } + \frac { 5 } { a _ { n } }\) for every positive integer \(n\).
Prove by mathematical induction that \(a _ { n } > 5\) for every positive integer \(n\).
Prove also that \(a _ { n } > a _ { n + 1 }\) for every positive integer \(n\).
3 The sequence $a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots$ is such that $a _ { 1 } > 5$ and $a _ { n + 1 } = \frac { 4 a _ { n } } { 5 } + \frac { 5 } { a _ { n } }$ for every positive integer $n$.\\
Prove by mathematical induction that $a _ { n } > 5$ for every positive integer $n$.
Prove also that $a _ { n } > a _ { n + 1 }$ for every positive integer $n$.
\hfill \mbox{\textit{CAIE FP1 2015 Q3}}