The Apple Method
Contents
What is the Apple Method?
The Apple Method is a method for solving algebra problems. An apple is used to make a clever algebraic substitution.
Why Apple?
A few reasons:
1. 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.
2. Apples are easier to draw.
3. Apples are good for you.
4. An Apple a Day Keeps the Doctor Away.
LaTeX code for apple
$(^{^(})$, or if you want some color, $\textcolor{red}{(\textcolor{green}{^{^(}})}$
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.
If we set 
equal to 
we get 
and 
Simplifying, we find 
so 
3. Evaluate: ![\[\frac{1^2+2^2+3^2+\cdots}{1^2+3^2+5^2+\cdots}\]](http://latex.artofproblemsolving.com/6/4/4/644c9ce7c476e8cd349b3561472a1ab51c3af623.png)
Extensions
The :) Method
When more than one variable is needed, pears, bananas, stars, and smiley faces are usually used.
