Chapter B.1, Problem 67E

If $A$ is $a$ positive real number, the terms pf the sequence defined by

$a_1=\frac{A}{2}\text{ and }{a}_n=\frac{1}{2}\left(a_{n-1}+\frac{A}{a_{n-1}}\right)\text{ for }n\ge 2$

can be used to approximate $\sqrt{A}$ to any decimal place accuracy desired. In Problems $67$ and $68$, compute the first four terms of this sequence for the indicated value of $A$, and compare the fourth term with the value of $\sqrt{A}$ obtained from a calculator.

$A=2$