2007 Specimen Paper P1 Q8

The positive numbers \({{x}_{n}}\) satisfy the relation \({{x}_{n+1}}=\sqrt{{{x}_{n}}+5}\) for \(n=1\), \(2\), \(3\), …

As \(n\to \infty \), \({{x}_{n}}\to l\).

1. Find (in either order) the value of \(l\) to \(3\) decimal places and the exact value of \(l\).
2. Prove that \({{\left( {{x}_{n+1}} \right)}^{2}}-{{l}^{2}}={{x}_{n}}-l\).
3. Hence show that, if \({{x}_{n}}>l\), then \({{x}_{n}}>{{x}_{n+1}}>l\).