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 $\sqrt{R}$.

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 $\sqrt{R}$.

`R=`

line, the for-loop, and the key interation formula in the `x=`

line.
Type your code here:

See your results here:

This code will not run! You have to fix the

`R=`

line. Set this equal to the number
you wish to take the square root of. Then, fix the for-loop to interate some number of times...10? 50? 100?
Lastly, program in the formula for the new x in the `x=`

line, from the current $x$ that exists
as this line is executed by the computer.
Note that the use of the assignment operator `=`

is quite clear here, as introducted
in a past lesson. We are suggesting
that you use `x`

on both sides of the `=`

sign. Is this OK?