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== | ||
− | |||
− | |||
− | If we set <math> | + | 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> | ||
+ | |||
+ | 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== | ||
− | + | ===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:
If we set , we can see that
.
Solving, we get
2. If
Find x.
If we set equal to
we get
and
Simplifying, we find so
3. Evaluate:
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.