Solve T_{n}=ar^n-1

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

Solve for a (complex solution)

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

Steps for Solving Linear Equation

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

$$ar^{n-1}=T_{n}$$

The equation is in standard form.

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

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

$$\frac{r^{n-1}a}{r^{n-1}}=\frac{T_{n}}{r^{n-1}}$$

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

$$a=\frac{T_{n}}{r^{n-1}}$$

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

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

Solve for a

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

Steps for Solving Linear Equation

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

$$ar^{n-1}=T_{n}$$

The equation is in standard form.

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

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

$$\frac{r^{n-1}a}{r^{n-1}}=\frac{T_{n}}{r^{n-1}}$$

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

$$a=\frac{T_{n}}{r^{n-1}}$$

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

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

Solve for T_n

$$T_{n}=ar^{n-1}$$

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