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, \(1.515980\ldots\)).
