$($Recursively$-$Defined Sequences$).($i$)$ Study the following program for a scientific calculator for

calculating the terms of the sequence $(p_n)$ recursively defined by

$p_n=p_{n-1}-f(p_{n-1})\frac{p_{n-1}-p_{n-2}}{f(p_{n-1})-f(p_{n-2})}$

for all $n2,$ where

$p_0=1$ and $p_1=0\mathrm{.}8$

and where the function $f$ is given by

$f(x)=xsin(0\mathrm{.}77x)-0\mathrm{.}39$

$($replace $p_0$ and $p_1$ in the program with the corresponding values given above, and replace $f(x)$ and

$f(Y)$ with correct expressions representing the values of the function $f$ given above; say, $f(x)$

should be replaced with

$xsin(0\mathrm{.}77x)-0\mathrm{.}39$

Prior to studying the above program, it is a good idea to start with creating your own program for calculating the terms $p_n$ in a

familiar programming language, and$/$or in Excel $($OpenOffice$),$ and then to compare your program with the program for a sci

calculator.

$($ii$)$ Execute the program for a sci calculator to get the terms $p_2,p_3,$dots,$p_7$ of the sequence $(p_n)$

$($make certain that your calculator is in RADIAN mode before commencing calculations$).$ Show your

work by filling in the following table; prior to entering in the table, all numerical values should be

rounded to $7-$digit floating$-$point numbers:

$($iii$)$ It can be shown that in fact the sequence $(p_n),$ generated by the so$-$called Secant Method for root

finding problems, converges to the unique root $$ of the function $f$ in $0,1.$ With this in mind, use the

WolframAlpha $(W|)$ website to find the $10-$digit approximation $\frac{?}{\text{b}\text{a}\text{r}}()$ of $$ by executing the command

solve $x**sin(0\mathrm{.}77**x)=0\mathrm{.}39in0,1$

at the website $($click on the $\frac{?}{\text{b}\text{a}\text{r}}()\frac{p_7}{b}ar()$