(i)[2]
Using a suitable pair of sketches, demonstrate that the equation $3e^x = 8 - 2x$ has a single root.
(ii)[2]
Confirm by calculation that this root is between $x = 0.7$ and $x = 0.8$.
(iii)[1]
Demonstrate that this root also meets the equation $x = \ln\left(\frac{8 - 2x}{3}\right)$.
(iv)[3]
Use the iterative formula $x_{n+1} = \ln\left(\frac{8 - 2x_n}{3}\right)$ to find this root correct to $3$ decimal places. Record each iteration to $5$ decimal places.