User:Delmont43/Recursive equations

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

1. Given:

   ${\displaystyle x=.{\overline {999}}}$

2. Multiply both sides by 10:

   ${\displaystyle 10x=9.{\overline {999}}}$

3. Subtract equation 1 from equation 2.:

   ${\displaystyle 10x-x=9.{\overline {999}}-.{\overline {999}}}$

4. Simplify:

   ${\displaystyle x=1\,}$

1. Given:

   ${\displaystyle {\sqrt {x-{\sqrt {x-{\sqrt {x-\cdots }}}}}}=4}$

2. Square both sides:

   ${\displaystyle \left({\sqrt {x-{\sqrt {x-{\sqrt {x-\cdots }}}}}}\right)^{2}=(4)^{2}}$

3. Simplify:

   ${\displaystyle x-{\sqrt {x-{\sqrt {x-{\sqrt {x-\cdots }}}}}}=16}$

4. Add equation 1 to equation 3.:

   ${\displaystyle \left(x-{\sqrt {x-{\sqrt {x-{\sqrt {x-\cdots }}}}}}\right)+\left({\sqrt {x-{\sqrt {x-{\sqrt {x-\cdots }}}}}}\right)=\left(16+4\right)}$

5. Simplify:

   ${\displaystyle x=20\,}$

Generalizing:

1. Given:

   ${\displaystyle y={\sqrt {x-{\sqrt {x-{\sqrt {x-\cdots }}}}}}}$

2. Square both sides:

   ${\displaystyle (y)^{2}=\left({\sqrt {x-{\sqrt {x-{\sqrt {x-\cdots }}}}}}\right)^{2}}$

3. Simplify:

   ${\displaystyle y^{2}=x-{\sqrt {x-{\sqrt {x-{\sqrt {x-\cdots }}}}}}}$

4. Add equation 1 to equation 3.:

   ${\displaystyle \left(y^{2}+y\right)=\left(x-{\sqrt {x-{\sqrt {x-{\sqrt {x-\cdots }}}}}}\right)+\left({\sqrt {x-{\sqrt {x-{\sqrt {x-\cdots }}}}}}\right)}$

5. Simplify:

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