Confusion with the formula to sum the terms of a finite geometric series The formula to sum the terms of a finite geometric series is the following:
$$\frac{a_1(1 - r^{n+1})}{1 - r}$$
where $a_1$ is the first term, $r$ is the common ratio, and $n + 1$ is the number we want to sum up to.
Now, my problem is really in this last part, I have seen some formulas that use $r^n$, others use $r^{n+1}$. 
My questions are:


*

*In general, how does the power to which we raise $r$ change depending on the index of the sum?

*What if we want to start the sum from a different index, for example $1$ or $3$ instead of $0$.

*How does the number of terms we want to sum influence the formula?
I know these might seem like stupid questions, but I am just confused, and it might be the time to understand exactly what's going on.
I have seen the derivation of the formula, but I am still not understanding the indices.

In general, for $n\geq m$ and $r\neq 1$,
$$
\sum_{k=m}^n r^k = r^m \sum_{k=m}^n r^{k-m}
= r^m \sum_{\ell=0}^{n-m} r^{\ell} = r^m\cdot \frac{1-r^{n-m+1}}{1-r}
$$
To remember it: the exponent in the denominator is the number of terms in the sum. 
To check (very basic check): when $n=m$, you only have one term, and it's $r^m$—so the result should simplify to $r^m$ in that case.
