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.
