The Apple Method

Revision as of 13:26, 17 May 2020 by Aops81619 (talk | contribs)

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 $x$ 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

1. Evaluate: \[\sqrt{6+\sqrt{6+\sqrt{6+\cdots}}}\]

$\emph{Solution:}$

If we set $\textcolor{red}{(\textcolor{green}{^{^(}})}=\sqrt{6+\sqrt{6+\sqrt{6+\cdots}}}$, we can see that $\textcolor{red}{(\textcolor{green}{^{^(}})}^2= 6+\textcolor{red}{(\textcolor{green}{^{^(}})}$.

Solving, we get $\textcolor{red}{(\textcolor{green}{^{^(}})}=\boxed{3}$

2. If \[\sqrt{x\cdot\sqrt{x\cdot\sqrt{x\cdots}}} = 5\]Find x.

3. Evaluate: \[\frac{1^2+2^2+3^2+\cdots}{1^2+3^3+5^2+\cdots}\]

Extensions

The pear method

When more than one variable is needed, pears, bananas, etc. are usually used.