Answer
See below.
Work Step by Step
Using a calculator:
$a_0=4$
$a_1=\frac{1}{2}(a_0+\frac{21}{a_0})=4.625$
$a_2=\frac{1}{2}(a_1+\frac{21}{a_1})\approx4.58277$
$a_3=\frac{1}{2}(a_2+\frac{21}{a_2})\approx4.58257$
$a_4=\frac{1}{2}(a_3+\frac{21}{a_3})\approx4.582575$
$a_5=\frac{1}{2}(a_4+\frac{21}{a_4})\approx4.5825757$
$\sqrt{21}=4.58257569496
$, so we got pretty close.