Geometric Progression prove the answer

Question

Geometric sequence has first term $a$ and last term $l$ and the sum of all these terms is $S$. Prove that the common ratio of the sequence is $\frac{S-a}{S-l}$.

How to include the answer with $S$?

Answer 1

Hint:

$$l=ar^{n-1}$$ $$ar^n=lr$$

Subsitute the above equation into $$S=\frac{a(r^n-1)}{r-1}$$

We obtain

$$S=\frac{lr-a}{r-1}$$

Solve for $r$

Answer 2

It is: $$\begin{align}S&=a+ar+ar^2+\cdots +ar^{n-2}+\underbrace{ar^{n-1}}_{l} \stackrel{\text{multiply by $r$}}{\Rightarrow}\\ rS&=ar+ar^2+\cdots +ar^{n-1}+lr \Rightarrow \\ rS&=S-a+lr \Rightarrow \\ r&=\frac{S-a}{S-l}.\end{align}$$