Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Difference between revisions of "The Apple Method"

(→‎Examples)
(→‎Examples)
 
(35 intermediate revisions by 8 users not shown)
Line 1: Line 1:
 +
==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?==
 +
A few reasons:
 +
 +
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.
 +
 +
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==
 
==Examples==
−
Evaluate: <cmath>\sqrt{6+\sqrt{6+\sqrt{6+\cdots}}}</cmath>
+
 
 +
1. Evaluate: <cmath>\sqrt{6+\sqrt{6+\sqrt{6+\cdots}}}</cmath>
 +
 
 +
<math>\emph{Solution:}</math>
 +
 
 +
If we set <math>\textcolor{red}{(\textcolor{green}{^{^(}})}=\sqrt{6+\sqrt{6+\sqrt{6+\cdots}}}</math>, we can see that <math>\textcolor{red}{(\textcolor{green}{^{^(}})}^2= 6+\textcolor{red}{(\textcolor{green}{^{^(}})}</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.
  
 
<math>\emph{Solution:}</math>
 
<math>\emph{Solution:}</math>
  
−
If we set <math>apple = \sqrt{6+\sqrt{6+\sqrt{6+\cdots}}}</math>, we can see that <math>apple = \sqrt{6+apple}</math>.
+
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>
 +
 
 +
Simplifying, we find <math>\textcolor{red}{(\textcolor{green}{^{^(}})} = x,</math> so <math>x = \boxed{5}</math>
 +
 
 +
3. Evaluate:
 +
<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>
 +
 
 +
<math>\emph{Solution:}</math>
 +
 
 +
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>
 +
 
 +
Thus, the answer is <math>\frac{\textcolor{red}{(\textcolor{green}{^{^(}})}}{\frac34\cdot\textcolor{red}{(\textcolor{green}{^{^(}})}}=\boxed{\frac34}.</math>
 +
 
 +
==Extensions==
  
−
Solving, we get <math>\boxed{apple = 3}</math>
+
===The :) Method===
 +
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.

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=The_Apple_Method&oldid=180436"