Difference between revisions of "The Apple Method"

Revision as of 12:57, 8 August 2022

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.

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 (link unavailable).

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.

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: $\sqrt{6+\sqrt{6+\sqrt{6+\cdots}}}$

$\emph{Solution:}$

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.

$\emph{Solution:}$

If we set $\sqrt{x\cdot\sqrt{x\cdot\sqrt{x\cdots}}}$ equal to $\textcolor{red}{(\textcolor{green}{^{^(}})},$ we get $\textcolor{red}{(\textcolor{green}{^{^(}})} = 5$ and $\textcolor{red}{(\textcolor{green}{^{^(}})}^2 = x \cdot \textcolor{red}{(\textcolor{green}{^{^(}})} = 25.$

Simplifying, we find $\textcolor{red}{(\textcolor{green}{^{^(}})} = x,$ so $x = \boxed{5}$

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

Extensions

The :) Method

When more than one variable is needed, pears, bananas, stars, and smiley faces are usually used.

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