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, \(1.515980\ldots\)).