Square root algorithm.

a$.$ Let $0=2$ and define $+1=$

$$

$2$

$+$

$1$

$$

for $=0,1,2,....$ Create a table showing $$

versus $$ for $=0,1,2,3,4,5.$ Notice the rapid convergence towards $2.$

b$.$ Let $0=3$ and define $+1=$

$$

$2$

$+$

$3$

$2$

for $=0,1,2,....$ Create a table showing $$

versus $$ for $=0,1,2,3,4,5.$ Notice the rapid convergence towards $3.$