Chapter 4.8, Problem 29E

(a) Apply Newton’s method to the equation x² − a = 0 to derive the following square-root algorithm (used by the ancient Babylonians to compute $\sqrt{a}$):

$x_{n+1}=\frac{1}{2}\left(x_n+\frac{a}{x_n}\right)$

(b) Use part (a) to compute $\sqrt{1000}$ correct to six decimal places.