By drawing appropriate graphs, show that the equation $e^{2x} = 6 + e^{-x}$ has exactly one real root.
Check by calculation that this root is between $0.5$ and $1$.
Show that, if a sequence defined by the iterative formula $x_{n+1} = \frac{1}{3}\ln(1 + 6e^{x_n})$ converges, then its limit is the root of the equation in part (i).
Use the iterative formula to find the root correct to $3$ decimal places. Show each iteration to $5$ decimal places.
Show that if a sequence of values defined by the iterative formula $x_{n+1} = \frac{1}{3}\ln\left(1 + 6e^{x_n}\right)$ converges, then its limit is the root of the equation in part (i).
Use the iterative formula to calculate the root correct to 3 decimal places. Show the outcome of each iteration to 5 decimal places.