Difference between revisions of "The Apple Method"
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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 | + | Simplifying, we find <math>\textcolor{red}{(\textcolor{green}{^{^(}})} = x,</math> so <math>x = \boxed{5}</math> |
3. Evaluate: <cmath>\frac{1^2+2^2+3^2+\cdots}{1^2+3^2+5^2+\cdots}</cmath> | 3. Evaluate: <cmath>\frac{1^2+2^2+3^2+\cdots}{1^2+3^2+5^2+\cdots}</cmath> | ||
Revision as of 01:38, 20 February 2021
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.
Dr. Ali Gurel from Alphastar academy started a new series of cool videos; the apple method's corresponding video can be found at https://www.youtube.com/watch?v=rz86M2hlOGk
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
$(^{^(})$
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: ![\[\frac{1^2+2^2+3^2+\cdots}{1^2+3^2+5^2+\cdots}\]](http://latex.artofproblemsolving.com/6/4/4/644c9ce7c476e8cd349b3561472a1ab51c3af623.png)
Extensions
The pear method
When more than one variable is needed, pears, bananas, and smiley faces are usually used.
