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?== | ||
| + | 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== | ||
| + | |||
1. Evaluate: <cmath>\sqrt{6+\sqrt{6+\sqrt{6+\cdots}}}</cmath> | 1. Evaluate: <cmath>\sqrt{6+\sqrt{6+\sqrt{6+\cdots}}}</cmath> | ||
| Line 10: | Line 28: | ||
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> | + | 2. If <cmath>\sqrt{x\cdot\sqrt{x\cdot\sqrt{x\cdots}}} = 5</cmath> |
| − | 3. Evaluate: <cmath>\frac{1^2+2^2+3^2+\cdots}{1^2+3^3+ | + | Find x. |
| + | |||
| + | <math>\emph{Solution:}</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== | ==Extensions== | ||
| − | ===The | + | |
| − | When more than one variable is needed, pears, bananas, | + | ===The :) Method=== |
| − | + | When more than one variable is needed, pears, bananas, stars, and smiley faces are usually used. | |
| − | |||
Latest revision as of 12:56, 8 November 2022
Contents
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 
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: ![\[\sqrt{6+\sqrt{6+\sqrt{6+\cdots}}}\]](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.
If we set 
equal to 
we get 
and 
Simplifying, we find 
so 
3. Evaluate:
![\[\frac{\frac{1}{1^2}+\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+\ldots}{\frac{1}{1^2}+\frac{1}{3^2}+\frac{1}{5^2}+\frac{1}{7^2}+\ldots}\]](http://latex.artofproblemsolving.com/9/2/0/920051af6280c49ad1415dedcb9531330fd5d18e.png)
Let 
. Note that 
Thus, the answer is 
Extensions
The :) Method
When more than one variable is needed, pears, bananas, stars, and smiley faces are usually used.
