What is the difference between these two geometric formulas?

Question

I've seen these two different geometric sum formulas but I don't know when which is used.

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

and

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

Answer 1

The second one is the sum for one additional term.

Note that

$\begin{array}\\ a\left(\frac{1-(r^{n+1})}{1-r}\right)-a\left(\frac{1-(r^{n})}{1-r}\right) &=a\left(\frac{(1-(r^{n+1}))-(1-(r^{n}))}{1-r}\right)\\ &=a\left(\frac{r^n-r^{n+1}}{1-r}\right)\\ &=a\left(\frac{r^n(1-r)}{1-r}\right)\\ &=ar^n \end{array} $

and this is the next term that is added to the first sum to get the second sum.