2016 MJC P1 Q6

The diagram below shows the graph of \({y = 2\ln(x - 1) + 4 - x}\).

The two roots of the equation \({2\ln(x - 1) + 4 - x = 0}\) are denoted by \({\alpha}\) and \({\beta}\), where ${\alpha

1. Find the values of \({\alpha}\) and \({\beta}\), correct to \(3\) decimal places.[2]

A sequence of real numbers \({x_1, x_2, x_3, \dots}\) where \({x_n > 1}\), satisfies the recurrence relation

\({{{x}_{n+1}}=\ln {{\left( {{x}_{n}}-1 \right)}^{2}}+4}\) for \({n\ge 1}\).

2. Prove algebraically that if the sequence converges, it must converge to either \({\alpha}\) or \({\beta}\).[2]
3. Use a calculator to determine the behaviour of the sequence for each of the cases \({x_1 = 3}\), \({x_1 = 12}\).[2]
4. By considering \({x_{n+1} - x_n}\) and the graph above, prove that
   \({{{x}_{n+1}}>{{x}_{n}}}\) if \({\alpha <{{x}_{n}}<\beta }\), \({{{x}_{n+1}}<{{x}_{n}}}\) if \(1<{{x}_{n}}<\alpha \) or \({{x}_{n}}>\beta \)[2]