Problem $4.-$ We can use an iterative method to calculate the square root of any number This method specifies that for any positive number $S,$ its square root $)=(\sqrt[\phantom{2}]{S}$ can be approximated by applying the following procedure.

Give an initial guess for the root, $r_0=$ Initial Guess $($any positive integer$)$

The next approximation is obtained using the following iterative formula.

$r_{i+1}=\frac{1}{2}(r_i+\frac{s}{r_i}),$ where $i=0,$dots,$n$

Repeat step $2$ until $,r_{i+1}=r_i$ or $(r_i-1-r_i),=0\mathrm{.}001$

Questions

Demonstrate that the above method can be used to calculate correctly the $\sqrt[\phantom{2}]{987}.$ This means $S=987.$ Assume that the initial guess $r_0=10.$ show all your calculations and fill out the following table as you calculate every iteration.

Table

$\backslash$table$[[r_0=,10],[r_1=,],[r_2=,],[r_3=,],[r_4=,],[r_5=,],[r_6=,],[r_7=,],[r_8=,],[r_9=,],[r_{10}=,],[,]]$

How many iterations or repetitions are needed to find the correct result with an error $0\mathrm{.}001$

Number of Iterations

Draw the flow diagram that can be used to describe the implementation of the iterative in any computer language, assuming that the user should provide the values of $S,r_0$ and $.$

Use Tinkercad to implement the program using an Arduino and demonstrate that it runs and obtains the correct answer.