An approximate solution to an equation is found using this iterative process.
\[
x_{n+1}=\frac{\left(x_{n}\right)^{3}-5}{10} \text { and } x_{1}=-1
\]
a) (i) Work out the value of $x_{2} \quad-0.6$
(ii) Work out the value of $x_{3}$
b) Work out the solution to 6 decimal places.
Answer
Iterate the formula until the difference between consecutive values is less than 10^-6: \(x_{n+1} = \frac{(x_{n})^{3} - 5}{10}\), Final Answer: \(\boxed{-0.513544}\)
Step 1 ：Find the value of x2 using the given formula and x1 = -1: \(x_{2} = \frac{(-1)^{3} - 5}{10} = -0.6\)
Step 2 ：Find the value of x3 using the same formula and x2: \(x_{3} = \frac{(-0.6)^{3} - 5}{10} = -0.5216\)
Step 3 ：Iterate the formula until the difference between consecutive values is less than 10^-6: \(x_{n+1} = \frac{(x_{n})^{3} - 5}{10}\), Final Answer: \(\boxed{-0.513544}\)
