Difference between revisions of "The Apple Method"
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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. | ||
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==Why Apple?== | ==Why Apple?== | ||
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==LaTeX code for apple== | ==LaTeX code for apple== | ||
− | $(^{^(})$ | + | $(^{^(})$, or if you want some color, $\textcolor{red}{(\textcolor{green}{^{^(}})}$ |
==Examples== | ==Examples== | ||
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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> | 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 = 5.</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> | ||
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+ | <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 | + | ===The :) Method=== |
− | When more than one variable is needed, pears, bananas, and smiley faces are usually used. | + | 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
Contents
[hide]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.