User:Delmont43/Recursive equations

From Wikipedia, the free encyclopedia

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

[edit] Example 1

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

[edit] Example 2

  1. Given:
    \sqrt{x - \sqrt{x - \sqrt{x - \cdots}}} = 4
  2. Square both sides:
    \left ( \sqrt{x - \sqrt{x - \sqrt{x - \cdots}}} \right ) ^2 = (4)^2
  3. Simplify:
    x - \sqrt{x - \sqrt{x - \sqrt{x - \cdots}}} = 16
  4. 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 )
  5. Simplify:
    x = 20 \,

Generalizing:

  1. Given:
    y = \sqrt{x - \sqrt{x - \sqrt{x - \cdots}}}
  2. Square both sides:
    (y)^2 = \left ( \sqrt{x - \sqrt{x - \sqrt{x - \cdots}}} \right ) ^2
  3. Simplify:
    y^2 = x - \sqrt{x - \sqrt{x - \sqrt{x - \cdots}}}
  4. 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 )
  5. Simplify:
    y^2 + y = x \,