# Iterative method

To iterate is to perform the same procedure over and over again. Sometimes we can solve equations approximately by iteration.

For example, to find an approximate solution to $x^3 + x = 5$, we can rewrite this as $x = \sqrt[3]{5-x}.$ Letting $x_{n+1} = \sqrt[3]{5-x_n}$ and taking our initial approximation to be $x_0 = 1$, we find \begin{align*} x_1 &= 1.58740105\ldots, \\ x_2 &= 1.50554966\ldots, \\ x_3 &= 1.51749158\ldots, \\ x_4 &= 1.51576099\ldots. \end{align*}

We can see that the values of $x_n$ appear to be approaching some number, which is a solution of the equation (in this case, $1.515980\ldots$).