= Finite geometric series
A finite geometric series is a <sum> of consecutive terms of a <geometric progression>. With first term $a$ and $N$ terms, including $N=0$ for the empty sum,
$$
\boxed{S_N=\sum_{j=0}^{N-1}ar^j=
\begin{cases}
a(1-r^N)/(1-r),&r\ne1,\\
Na,&r=1.
\end{cases}}
$$
Unlike an infinite <geometric series>, this identity does not need $|r|<1$. For example, $3+6+12+24+48=3(2^5-1)=93$. The number of terms is five; the largest exponent is four.
The cancellation behind the formula is especially useful to remember: multiplying $S_N$ by the <common ratio> shifts all terms one place. In $S_N-rS_N$, every interior term cancels, leaving $a-ar^N$. Division by $1-r$ gives the identity when $r\ne1$; at $r=1$ the sum is directly $Na$. The same argument works for <complex numbers>.
For integers $m\le n$ and $r\ne1$, the shifted version is
$$
\sum_{j=m}^n r^j=r^m\frac{1-r^{n-m+1}}{1-r}.
$$
There are $n-m+1$ terms. All powers must be defined; if a negative exponent is present, $r$ must be nonzero. At $r=1$ this sum is simply $n-m+1$. The <finite trigonometric sum> uses this identity with a <complex exponential> as the <common ratio>.
Finite sums of <positive integers> in a <geometric progression> already appear in <Euclid's Elements>, Book IX, Proposition 35. The argument there compares differences of proportional numbers; Joyce's guide rewrites its conclusion as the modern finite-sum formula. This historical result concerns finite sums, without asserting an infinite <limit>. https://mathcs.clarku.edu/~djoyce/elements/bookIX/propIX35.html[Euclid, Elements IX.35, with David Joyce's guide to the algebraic form]{ref}
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