Answer
$\approx 2.828$
Work Step by Step
We have to determine an approximation of $\sqrt 8$.
We start with an approximation:
$a_0=2$
Compute $a_1,a_2,a_3,a_4,a_5$ using the formula:
$a_n=\dfrac{1}{2}\left(a_{n-1}+\dfrac{p}{a_{n-1}}\right)$, where $p=5$.
$a_1=\dfrac{1}{2}\left(a_0+\dfrac{8}{a_0}\right)=\dfrac{1}{2}\left(2+\dfrac{8}{2}\right)=3$
$a_2=\dfrac{1}{2}\left(a_1+\dfrac{5}{a_1}\right)=\dfrac{1}{2}\left(3+\dfrac{8}{3}\right)\approx 2.8333333$
$a_3=\dfrac{1}{2}\left(a_2+\dfrac{8}{a_2}\right)=\dfrac{1}{2}\left(2.8333333+\dfrac{8}{2.8333333}\right)\approx 2.8284314$
$a_4=\dfrac{1}{2}\left(a_3+\dfrac{8}{a_3}\right)=\dfrac{1}{2}\left(2.8284314+\dfrac{8}{2.8284314}\right)\approx 2.8284271$
$a_5=\dfrac{1}{2}\left(a_4+\dfrac{8}{a_4}\right)=\dfrac{1}{2}\left(2.8284271+\dfrac{8}{2.8284271}\right)\approx 2.8284271$
So $\sqrt 8\approx 2.828$
