$($a$)$ x Calculate r by hand and write down the result as an analytical formula $($in terms of

k$).$

$($b$)$ Using the formula found in $($a$),$ calculate the value of r numerically $($meaning$,$ using

MATLAB$)$ for k $2,3,$ and $4.$

$($c$)$ Let r$00$ and

r$1?$k $$ ak $`$ r$0$

r$2$

b

k $`?$k $$ ak $`$ r$1$

r$3$

c

k $`$

b

k $`?$k $$ ak $`$ r$2$

$...$

So$,$ for any n P N$,$ rn $?$k $`$ rn $1$ and r $$ limn$8$ rn$.$ Thus r can be approximated by

rn for sufficiently large n$.$ Now, for k $7,$ determine the smallest integer n such that

$|$r $$ rn$|108$