Solve S=a(1-r^n)/1-r

$$S = \frac { a ( 1 - r ^ { n } ) } { 1 - r }$$

Solve for a

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

Steps for Solving Linear Equation

Multiply both sides of the equation by -r+1.

$$S\left(-r+1\right)=a\left(1-r^{n}\right)$$

Use the distributive property to multiply S by -r+1.

$$-Sr+S=a\left(1-r^{n}\right)$$

Use the distributive property to multiply a by 1-r^{n}.

$$-Sr+S=a-ar^{n}$$

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

$$a-ar^{n}=-Sr+S$$

Combine all terms containing a.

$$\left(1-r^{n}\right)a=-Sr+S$$

The equation is in standard form.

$$\left(1-r^{n}\right)a=S-Sr$$

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

$$\frac{\left(1-r^{n}\right)a}{1-r^{n}}=\frac{S-Sr}{1-r^{n}}$$

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

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

Divide -Sr+S by 1-r^{n}.

$$a=\frac{S\left(1-r\right)}{1-r^{n}}$$

Solve for S

$$S=-\frac{a\left(r^{n}-1\right)}{1-r}$$

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