9231 P11 - Jun 2015 - Q3 - 7 marks
6306
The sequence \(a_{1}, a_{2}, a_{3}, \ldots\) is such that \(a_{1}\gt 5\) and \(a_{n+1}=\frac{4 a_{n}}{5}+\frac{5}{a_{n}}\) for every positive integer \(n\). Prove by mathematical induction that \(a_{n}\gt 5\) for every positive integer \(n\).
Prove also that \(a_{n}\gt a_{n+1}\) for every positive integer \(n\).
