Sum of a Geometric Series

What is the sum of the following geometric series?
We will frequently need a simple formula for this finite series. It is called a geometric series because each term is related by a multiple to the previous one. If each term was related by a fixed difference, it would be called an arithmetic series; but that’s for another day.
Let us define the sum as \(S\). Then writing the equation for \(S\) and \(aS\) with some clever alignment:

S=&1&+a&+a^2&+…&+a^k \\
Subtract the equations
\frac{1-a^{k+1}}{1-a} &a \neq 1 \\
k+1 &\text{otherwise}
where we have to be careful to divide by \(1-a\) only if \(a\neq 1\), and the answer for \(a=1\) is determined by inspection.

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