FIXED POINT ALGORITHM.
If $g$ is a continuous function taking the interval $[a,b]$ to itself, then it has a fixed point $r\in [a,b]$ so that $r = g(r).$ If in addition, $g$ is differentiable and satisfies $| g'(x)|\le M$ for all $a \le x\le b$ where $M<1$ is a constant, then the recursion $x_{n+1}=g(x_n),\ x_1\in [a,b]$ yields a sequence that converges $x_n\to r$ as $n\to\infty.$

Consider the equation $x= \sqrt{ 6 + x }.$ Using the Fixed Point Algorithm starting with $x_{1} = 2$, find $x_{2}$ to $x_{7}$.

$x_{1} = 2$
$x_{2} =$
$x_{3} =$
$x_{4} =$
$x_{5} =$
$x_{6} =$
$x_{7} =$

Solve for the (positive) $x$ in $x=\sqrt{6 + x}.$
$x=$

Evaluate $\sqrt{6+\sqrt{6+\sqrt{6+\cdots}}} .$