In a past lesson you used a for-loop to iterate a function over and over again. In that case, you looked for a given pattern to emerge. Here's a nifty iteration that will end up giving you the square root of a number (see Cheney, et. al., 5th ed, p. 117).

Suppose you want to find the square root of a number in the variable $R$, and you start with a guess in varable $x$. Perhaps $x$ can start at $R/2$ (since you know the square root of a number is always less than the number itself). We'll call this initial $x$, $x_0$. You can find a still better approximation to $\sqrt{R}$ by finding the next $x$, or $x_1$ by: $$x_1=\frac{1}{2}(x_0+\frac{R}{x_0})$$ In general, the iteration to find $\sqrt{R}$ is given by $$x_{n+1}=\frac{1}{2}(x_n+\frac{R}{x_n})$$

In this lesson, try to practice your for-loops and programming formulas into the computer by writing some code to find $\sqrt{R}$.