Answer
See below.
Work Step by Step
Using a calculator:
$a_0=2$
$a_1=\frac{1}{2}(a_0+\frac{5}{a_0})\approx2.25$
$a_2=\frac{1}{2}(a_1+\frac{5}{a_1})\approx2.23611$
$a_3=\frac{1}{2}(a_2+\frac{5}{a_2})\approx2.23606$
$a_4=\frac{1}{2}(a_3+\frac{5}{a_3})\approx2.2423$
$a_5=\frac{1}{2}(a_4+\frac{5}{a_4})\approx2.236$
$\sqrt{5}=2.2360679775$, so we got pretty close.