Question $4(3$ points$)$

In class we discussed Heron's Method and how to derive it from Newton's method.

Derive a recurrence relation using Newton's Method which iteratively calculates

$(\backslash$sqrt$($x$))/(7)$

What is the recurrence relation?

a$\_($n$)=$a$\_($n$-1)-(49)/(2)($a$\_($n$-1)-($x$)/($a$\_($n$-1)))$

a$\_($n$)=$a$\_($n$-1)-(1)/(2)($a$\_($n$-1)-($x$)/(49$a$\_($n$-1)))$

a$\_($n$)=$a$\_($n$-1)-(1)/(2)($a$\_($n$-1)-($x$)/($a$\_($n$-1)))$

a$\_($n$)=$a$\_($n$-1)-(1)/(2)($a$\_($n$-1)-(7$x$)/($a$\_($n$-1)))$In class we discussed Heron's Method and how to derive it from Newton's method.

Derive a recurrence relation using Newton's Method which iteratively calculates

$\frac{\sqrt[\phantom{2}]{x}}{7}$

What is the recurrence relation?

$a_n=a_{n-1}-\frac{1}{2}(a_{n-1}-\frac{7x}{a_{n-1}})$

$a_n=a_{n-1}-\frac{49}{2}(a_{n-1}-\frac{x}{a_{n-1}})$

$a_n=a_{n-1}-\frac{1}{2}(a_{n-1}-\frac{x}{a_{n-1}})$

$a_n=a_{n-1}-\frac{1}{2}(a_{n-1}-\frac{x}{49a_{n-1}})$