By sketching a suitable pair of graphs, show that the equation \(\ln ( x + 2 ) = 4 \mathrm { e } ^ { - x }\) has exactly one real root.
Show by calculation that this root lies between \(x = 1\) and \(x = 1.5\).
Use the iterative formula \(x _ { n + 1 } = \ln \left( \frac { 4 } { \ln \left( x _ { n } + 2 \right) } \right)\) to determine the root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.