User:Delmont43/Recursive equations

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

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:

   $x=1\,$

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:

   $y^{2}+y=x\,$