Difference between revisions of "The Apple Method"
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Solving, we get <math>\textcolor{red}{(\textcolor{green}{^{^(}})}=\boxed{3}</math> | Solving, we get <math>\textcolor{red}{(\textcolor{green}{^{^(}})}=\boxed{3}</math> | ||
| − | 2. Evaluate: <cmath>\frac{1^2+2^2+3^2+\cdots}{1^2+3^3+5^2+\cdots}</cmath> | + | 2. If <cmath>\sqrt{x\cdot\sqrt{x\cdot\sqrt{x\cdots}}} = 5</cmath>Find x. |
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| + | 3. Evaluate: <cmath>\frac{1^2+2^2+3^2+\cdots}{1^2+3^3+5^2+\cdots}</cmath> | ||
Revision as of 22:36, 24 March 2020
The Apple Method is a method for solving algebra problems. An apple is used to make a clever algebraic substitution.
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.
3. Evaluate: ![\[\frac{1^2+2^2+3^2+\cdots}{1^2+3^3+5^2+\cdots}\]](http://latex.artofproblemsolving.com/5/1/e/51ebeab998a84de66e0d06d6d48891f9b18f1820.png)
