Answer
$\approx 4.5825757$
Work Step by Step
We apply the given recursive formula to find the approximate value for $\sqrt {21}$ as follows:
$$\displaystyle a_0=4 \\a_1=\frac{1}{2}(a_0+\frac{21}{a_0})=4.625 \\ a_2=\frac{1}{2}(a_1+\frac{21}{a_1})\approx4.58277 \\ a_3=\frac{1}{2}(a_2+\frac{21}{a_2})\approx4.58257 \\a_4=\frac{1}{2}(a_3+\frac{21}{a_3})\approx4.582575 \\ a_5=\frac{1}{2}(a_4+\frac{21}{a_4})\approx4.5825757$$
Therefore, we get the approximate value for
$\sqrt{21}=4.58257569496 \approx 4.5825757$
