The Apple Method
The Apple Method is a method for solving algebra problems. An apple is used to make a clever algebraic substitution.
Examples
1. Evaluate: ![\[\sqrt{6+\sqrt{6+\sqrt{6+\cdots}}}\]](http://latex.artofproblemsolving.com/b/a/9/ba931c757fe2ab64fb02b7b34de5a0e22ea0b44c.png)
If we set 
, we can see that 
.
Solving, we get 
2. If ![\[\sqrt{x\cdot\sqrt{x\cdot\sqrt{x\cdots}}} = 5\]](http://latex.artofproblemsolving.com/5/5/0/5500be6aeb9adf10d0b25bcdcbeea78f7b9c7098.png)
Find x.
3. Evaluate: ![\[\frac{1^2+2^2+3^2+\cdots}{1^2+3^3+5^2+\cdots}\]](http://latex.artofproblemsolving.com/5/1/e/51ebeab998a84de66e0d06d6d48891f9b18f1820.png)
Extensions
The pear method
When more than one variable is needed, pears, bananas, etc. are usually used.
Why Apple?
When you use the Apple Method, you can box what you are substituting with the apple. When you use 
as a substitution, instead of actually boxing it, you are just crossing it out.
