Solve (2+sqrt{2})^n=An+sqrt{2}Bn

$$( 2 + \sqrt { 2 } ) ^ { n } = A n + \sqrt { 2 } B n$$

Solve for A

$$A=\frac{\left(4\sqrt{2}+6\right)^{\frac{n}{2}}}{n}-\sqrt{2}B$$

$$n\neq 0$$

Steps for Solving Linear Equation

Swap sides so that all variable terms are on the left hand side.

$$An+\sqrt{2}Bn=\left(2+\sqrt{2}\right)^{n}$$

Subtract \sqrt{2}Bn from both sides.

$$An=\left(2+\sqrt{2}\right)^{n}-\sqrt{2}Bn$$

Reorder the terms.

$$An=\left(\sqrt{2}+2\right)^{n}-\sqrt{2}Bn$$

The equation is in standard form.

$$nA=\left(\sqrt{2}+2\right)^{n}-\sqrt{2}Bn$$

Divide both sides by n.

$$\frac{nA}{n}=\frac{\left(\sqrt{2}+2\right)^{n}-\sqrt{2}Bn}{n}$$

Dividing by n undoes the multiplication by n.

$$A=\frac{\left(\sqrt{2}+2\right)^{n}-\sqrt{2}Bn}{n}$$

Divide \left(\sqrt{2}+2\right)^{n}-\sqrt{2}Bn by n.

$$A=\frac{\left(\sqrt{2}+2\right)^{n}}{n}-\sqrt{2}B$$

Solve for B

$$B=-\frac{\sqrt{2}\left(-\left(2\left(2\sqrt{2}+3\right)\right)^{\frac{n}{2}}+An\right)}{2n}$$

$$n\neq 0$$

Steps for Solving Linear Equation

Swap sides so that all variable terms are on the left hand side.

$$An+\sqrt{2}Bn=\left(2+\sqrt{2}\right)^{n}$$

Subtract An from both sides.

$$\sqrt{2}Bn=\left(2+\sqrt{2}\right)^{n}-An$$

The equation is in standard form.

$$\sqrt{2}nB=\left(\sqrt{2}+2\right)^{n}-An$$

Divide both sides by \sqrt{2}n.

$$\frac{\sqrt{2}nB}{\sqrt{2}n}=\frac{\left(\sqrt{2}+2\right)^{n}-An}{\sqrt{2}n}$$

Dividing by \sqrt{2}n undoes the multiplication by \sqrt{2}n.

$$B=\frac{\left(\sqrt{2}+2\right)^{n}-An}{\sqrt{2}n}$$

Divide \left(2+\sqrt{2}\right)^{n}-An by \sqrt{2}n.

$$B=\frac{\sqrt{2}\left(\sqrt{2}+2\right)^{n}}{2n}-\frac{\sqrt{2}A}{2}$$