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What is the sum of the following geometric series?

\[

\sum_{i=0}^{k}a^i=1+a+a^2+…+a^k

\]

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:

\begin{array}{rrrccc}

S=&1&+a&+a^2&+…&+a^k \\

aS=&&a&+a^2&+…&+a^k&+a^{k+1}

\end{array}

Subtract the equations

\[

S(1-a)=1-a^{k+1}

\]

or

\begin{align}

S=

\begin{cases}

\frac{1-a^{k+1}}{1-a} &a \neq 1 \\

k+1 &\text{otherwise}

\end{cases}

\end{align}

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.

Q.E.D.