Chapter 4, Problem 8RWDT

To determine

To find : An old algorithm for estimating $\sqrt{N}$ is to make a guess, $G$ , and use the average of $G$ and $\frac{N}{G}$ as your next guess (and so on). Try this on your calculator, using 5 as your first guess for $\sqrt{29}$ . How many iterations are required to find $\sqrt{29}$ correctly to ten digits?

Answer to Problem 8RWDT

Three iterations are needed to find $\sqrt{29}$ correctly to 10 digits.

Explanation of Solution

Given : N = $29$ and our first guess (G) is $5$ .

First Iteration :

N/G = $\frac{29}{5}=5.8$

Average of N and N/G = $\frac{5+5.8}{2}=\frac{10.8}{2}=5.4$

Now, squaring $5.4$ , ${\left(5.4\right)}^2=29.16$

Second Iteration :

G = $5.4$ , N/G = $\frac{29}{5.8}=5$

Average of N and N/G = $\frac{5.4+5}{2}=\frac{10.4}{2}=5.2$

Now, squaring $5.2$ , ${\left(5.2\right)}^2=27.04$

Third Iteration :

G = 5.2, N/G = $\frac{29}{5.2}=5.5769230769230$

Average of N and N/G = $\frac{5.2+5.5769230769230}{2}=5.3884615384615$

Now, squaring $5.3884615384615$ , ${\left(5.3884615384615\right)}^2=29.035517751479$