2018 IJC MYE Q12 [Modified]

The diagram shows the graph of \(y=\ln \left( x+2 \right)-\frac{x}{5}\).

The positive root of the equation \(\ln \left( x+2 \right)-\frac{x}{5}=0\) is denoted by \(\alpha \).

Find the value of \(\alpha \).

A sequence of positive real numbers \({{x}_{1}}\), \({{x}_{2}}\), \({{x}_{3}}\), … satisfies the recurrence relation

\({{x}_{n+1}}=5\ln \left( {{x}_{n}}+2 \right)\) for \(n\ge 1\).

1. Prove algebraically that, if the sequence converges, then it converges to \(\alpha \).
2. Use a calculator to determine the behaviour of the sequence for each of the cases \({{x}_{1}}=1\) and \({{x}_{1}}=20\).
3. By considering \({{x}_{n+1}}-{{x}_{n}}\), prove that
   \({{x}_{n+1}}>{{x}_{n}}\) if \(0<{{x}_{n}}<\alpha \)
   \({{x}_{n+1}}<{{x}_{n}}\) if \({{x}_{n}}>\alpha \)