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.
Examples
Dr. Ali Gurel from Alphastar academy started a new series of cool videos; the apple method's corresponding video can be found at https://www.youtube.com/watch?v=rz86M2hlOGk , and the website for the series can be found at https://sites.google.com/view/cool-math-solutions/home.
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: ![\[\frac{1^2+2^2+3^2+\cdots}{1^2+3^2+5^2+\cdots}\]](http://latex.artofproblemsolving.com/6/4/4/644c9ce7c476e8cd349b3561472a1ab51c3af623.png)
Extensions
The pear method
When more than one variable is needed, pears, bananas, and smiley faces are usually used.
