User:Delmont43/Recursive equations

Recursive equations can often be solved using the technique shown below:

Example 1

 * 1) Given:
 * $$x = .\overline{999}$$
 * 1) Multiply both sides by 10:
 * $$10x = 9.\overline{999}$$
 * 1) Subtract equation 1 from equation 2.:
 * $$10x - x = 9.\overline{999} - .\overline{999}$$
 * 1) Simplify:
 * $$x = 1 \,$$

Example 2

 * 1) Given:
 * $$\sqrt{x - \sqrt{x - \sqrt{x - \cdots}}} = 4$$
 * 1) Square both sides:
 * $$\left ( \sqrt{x - \sqrt{x - \sqrt{x - \cdots}}} \right ) ^2 = (4)^2$$
 * 1) Simplify:
 * $$x - \sqrt{x - \sqrt{x - \sqrt{x - \cdots}}} = 16$$
 * 1) Add equation 1 to equation 3.:
 * $$\left ( x - \sqrt{x - \sqrt{x - \sqrt{x - \cdots}}} \right ) + \left ( \sqrt{x - \sqrt{x - \sqrt{x - \cdots}}} \right ) = \left ( 16 + 4 \right )$$
 * 1) Simplify:
 * $$x = 20 \,$$

Generalizing:


 * 1) Given:
 * $$y = \sqrt{x - \sqrt{x - \sqrt{x - \cdots}}}$$
 * 1) Square both sides:
 * $$(y)^2 = \left ( \sqrt{x - \sqrt{x - \sqrt{x - \cdots}}} \right ) ^2$$
 * 1) Simplify:
 * $$y^2 = x - \sqrt{x - \sqrt{x - \sqrt{x - \cdots}}}$$
 * 1) Add equation 1 to equation 3.:
 * $$\left ( y^2 + y \right ) = \left ( x - \sqrt{x - \sqrt{x - \sqrt{x - \cdots}}} \right ) + \left ( \sqrt{x - \sqrt{x - \sqrt{x - \cdots}}} \right )$$
 * 1) Simplify:
 * $$y^2 + y = x \,$$