The Apple Method

The Apple Method is a method for solving algebra problems. An apple is used to make a clever algebraic substitution.

Examples

1. Evaluate: $\sqrt{6+\sqrt{6+\sqrt{6+\cdots}}}$

$\emph{Solution:}$

If we set $\textcolor{red}{(\textcolor{green}{^{^(}})}=\sqrt{6+\sqrt{6+\sqrt{6+\cdots}}}$ , we can see that $\textcolor{red}{(\textcolor{green}{^{^(}})}^2= 6+\textcolor{red}{(\textcolor{green}{^{^(}})}$ .

Solving, we get $\textcolor{red}{(\textcolor{green}{^{^(}})}=\boxed{3}$

2. If $\sqrt{x\cdot\sqrt{x\cdot\sqrt{x\cdots}}} = 5$ Find x.

3. Evaluate: $\frac{1^2+2^2+3^2+\cdots}{1^2+3^3+5^2+\cdots}$

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The Apple Method

Examples