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.