Difference between revisions of "The Apple Method"

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==What is the Apple Method?==
 
The Apple Method is a method for solving algebra problems.
 
The Apple Method is a method for solving algebra problems.
 
An apple is used to make a clever algebraic substitution.
 
An apple is used to make a clever algebraic substitution.
  
 
==Why Apple?==
 
==Why Apple?==
A few reasons:\
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A few reasons:
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1. When you use the Apple Method, you can box what you are substituting with the apple. When you use <math>x</math> as a substitution, instead of actually boxing it, you are just crossing it out.
 
1. When you use the Apple Method, you can box what you are substituting with the apple. When you use <math>x</math> as a substitution, instead of actually boxing it, you are just crossing it out.
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2. Apples are easier to draw.
 
2. Apples are easier to draw.
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3. Apples are good for you.
 
3. Apples are good for you.
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4. An Apple a Day Keeps the Doctor Away.
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==LaTeX code for apple==
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$(^{^(})$, or if you want some color, $\textcolor{red}{(\textcolor{green}{^{^(}})}$
  
 
==Examples==
 
==Examples==
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1. Evaluate: <cmath>\sqrt{6+\sqrt{6+\sqrt{6+\cdots}}}</cmath>
 
1. Evaluate: <cmath>\sqrt{6+\sqrt{6+\sqrt{6+\cdots}}}</cmath>
  
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Solving, we get <math>\textcolor{red}{(\textcolor{green}{^{^(}})}=\boxed{3}</math>
 
Solving, we get <math>\textcolor{red}{(\textcolor{green}{^{^(}})}=\boxed{3}</math>
  
2. If <cmath>\sqrt{x\cdot\sqrt{x\cdot\sqrt{x\cdots}}} = 5</cmath>Find x.
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2. If <cmath>\sqrt{x\cdot\sqrt{x\cdot\sqrt{x\cdots}}} = 5</cmath>  
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Find x.
  
3. Evaluate: <cmath>\frac{1^2+2^2+3^2+\cdots}{1^2+3^3+5^2+\cdots}</cmath>
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<math>\emph{Solution:}</math>
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If we set <math>\sqrt{x\cdot\sqrt{x\cdot\sqrt{x\cdots}}}</math> equal to <math>\textcolor{red}{(\textcolor{green}{^{^(}})},</math> we get <math>\textcolor{red}{(\textcolor{green}{^{^(}})} = 5</math> and <math>\textcolor{red}{(\textcolor{green}{^{^(}})}^2 = x \cdot \textcolor{red}{(\textcolor{green}{^{^(}})} = 25.</math>
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Simplifying, we find <math>\textcolor{red}{(\textcolor{green}{^{^(}})} = x,</math> so <math>x = \boxed{5}</math>
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3. Evaluate:
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<cmath>\frac{\frac{1}{1^2}+\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+\ldots}{\frac{1}{1^2}+\frac{1}{3^2}+\frac{1}{5^2}+\frac{1}{7^2}+\ldots}</cmath>
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<math>\emph{Solution:}</math>
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Let <math>\textcolor{red}{(\textcolor{green}{^{^(}})}=\frac{1}{1^2}+\frac{1}{2^2}+\frac{1}{3^2}+\cdots</math>. Note that <math>\frac{1}{1^2}+\frac{1}{3^2}+\frac{1}{5^2}+\cdots = \left( \frac{1}{1^2}+\frac{1}{2^2}+\frac{1}{3^2}+\cdots \right) - \left( \frac{1}{2^2}+\frac{1}{4^2}+\frac{1}{6^2}+\cdots \right) = \textcolor{red}{(\textcolor{green}{^{^(}})} - \frac{1}{2^2}\cdot\textcolor{red}{(\textcolor{green}{^{^(}})} = \frac{3}{4}\cdot\textcolor{red}{(\textcolor{green}{^{^(}})}.</math>
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Thus, the answer is <math>\frac{\textcolor{red}{(\textcolor{green}{^{^(}})}}{\frac34\cdot\textcolor{red}{(\textcolor{green}{^{^(}})}}=\boxed{\frac34}.</math>
  
 
==Extensions==
 
==Extensions==
===The pear method===
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When more than one variable is needed, pears, bananas, etc. are usually used.
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===The :) Method===
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When more than one variable is needed, pears, bananas, stars, and smiley faces are usually used.

Latest revision as of 11:56, 8 November 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.

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: \[\frac{\frac{1}{1^2}+\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+\ldots}{\frac{1}{1^2}+\frac{1}{3^2}+\frac{1}{5^2}+\frac{1}{7^2}+\ldots}\]

$\emph{Solution:}$

Let $\textcolor{red}{(\textcolor{green}{^{^(}})}=\frac{1}{1^2}+\frac{1}{2^2}+\frac{1}{3^2}+\cdots$. Note that $\frac{1}{1^2}+\frac{1}{3^2}+\frac{1}{5^2}+\cdots = \left( \frac{1}{1^2}+\frac{1}{2^2}+\frac{1}{3^2}+\cdots \right) - \left( \frac{1}{2^2}+\frac{1}{4^2}+\frac{1}{6^2}+\cdots \right) = \textcolor{red}{(\textcolor{green}{^{^(}})} - \frac{1}{2^2}\cdot\textcolor{red}{(\textcolor{green}{^{^(}})} = \frac{3}{4}\cdot\textcolor{red}{(\textcolor{green}{^{^(}})}.$

Thus, the answer is $\frac{\textcolor{red}{(\textcolor{green}{^{^(}})}}{\frac34\cdot\textcolor{red}{(\textcolor{green}{^{^(}})}}=\boxed{\frac34}.$

Extensions

The :) Method

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