By sketching a suitable pair of graphs, show that the equation
$$5 \mathrm { e } ^ { - x } = \sqrt { } x$$
has one root.
Show that, if a sequence of values given by the iterative formula
$$x _ { n + 1 } = \frac { 1 } { 2 } \ln \left( \frac { 25 } { x _ { n } } \right)$$
converges, then it converges to the root of the equation in part (i).
Use this iterative formula, with initial value \(x _ { 1 } = 1\), to calculate the root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.