(a)[2]
Show by sketching an appropriate pair of graphs that the equation $x^5 = 2 + x$ has one and only one real root.
(b)[2]
Show that, if the sequence defined by the iterative formula $$x_{n+1} = \frac{4x_n^5 + 2}{5x_n^4 - 1}$$ does converge, its limit is the root of the equation in part (a).
(c)[3]
Use the iterative formula, starting from $x_1 = 1.5$, to determine the root correct to $3$ decimal places. State each iterated value to $5$ decimal places.