Solve a_{n}=a_{1}r^n-1

$$a _ { n } = a _ { 1 } r ^ { n - 1 }$$

Solve for a_1 (complex solution)

$$\left\{\begin{matrix}a_{1}=a_{n}r^{1-n}\text{, }&n=1\text{ or }r\neq 0\\a_{1}\in \mathrm{C}\text{, }&a_{n}=0\text{ and }r=0\text{ and }n\neq 1\end{matrix}\right.$$

Steps for Solving Linear Equation

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

$$a_{1}r^{n-1}=a_{n}$$

The equation is in standard form.

$$r^{n-1}a_{1}=a_{n}$$

Divide both sides by r^{n-1}.

$$\frac{r^{n-1}a_{1}}{r^{n-1}}=\frac{a_{n}}{r^{n-1}}$$

Dividing by r^{n-1} undoes the multiplication by r^{n-1}.

$$a_{1}=\frac{a_{n}}{r^{n-1}}$$

Divide a_{n} by r^{n-1}.

$$a_{1}=a_{n}r^{1-n}$$

Solve for a_1

$$\left\{\begin{matrix}a_{1}=a_{n}r^{1-n}\text{, }&\left(r<0\text{ and }Denominator(n)\text{bmod}2=1\right)\text{ or }r>0\\a_{1}\in \mathrm{R}\text{, }&a_{n}=0\text{ and }r=0\text{ and }n>1\end{matrix}\right.$$

Steps for Solving Linear Equation

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

$$a_{1}r^{n-1}=a_{n}$$

The equation is in standard form.

$$r^{n-1}a_{1}=a_{n}$$

Divide both sides by r^{n-1}.

$$\frac{r^{n-1}a_{1}}{r^{n-1}}=\frac{a_{n}}{r^{n-1}}$$

Dividing by r^{n-1} undoes the multiplication by r^{n-1}.

$$a_{1}=\frac{a_{n}}{r^{n-1}}$$

Divide a_{n} by r^{n-1}.

$$a_{1}=a_{n}r^{1-n}$$

Solve for a_n

$$a_{n}=a_{1}r^{n-1}$$

$$r>0\text{ or }\left(r=0\text{ and }n>1\right)\text{ or }\left(r<0\text{ and }Denominator(n)\text{bmod}2=1\right)$$