Proof

Sum of a Geometric Series

What follows is a derivation of the formula for the sum of a geometric series.
Consider a geometric series of

n

terms, whose first term is

a_{1}

and whose common ratio is

r .

The sum of the series can be written as

S_{n} = a_{1} + a_{1} r + a_{1} r^{2} + a_{1} r^{3} + \dots + a_{1} r^{n - 1}

Notice that the series is a polynomial, and can be written in standard form.

S_{n} = a_{1} r^{n - 1} + \dots + a_{1} r^{3} + a_{1} r^{2} + a_{1} r + a

Since all of the terms contain a factor of

a_{1},

it can be factored out of the right-hand side.

S_{n} = a_{1} (r^{n - 1} + \dots + r^{3} + r^{2} + r + 1)

Since the sum above is a polynomial, polynomial division can be used to rewrite it. In fact, one polynomial identity gives

\frac{1 - r ^{n}}{1 - r} = r^{n - 1} + \dots + r^{3} + r^{2} + r + 1 .

Thus, the formula for the sum of a geometric series can be written as follows.

$S_{n} = \frac{a _{1} ( 1 - r ^{n} )}{1 - r}$

Q.E.D.