The formula for the *n*-th partial sum, S_{n}, of a geometric series with common ratio *r* is given by:

This formula is actually quite simple to confirm: you just use polynomial long division.

The sum of the first *n* terms of the geometric sequence, in expanded form, is as follows:

*a* + *ar* + *ar*^{2} + *ar*^{3} + ... + *ar ^{n}*

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Polynomials are customarily written with their terms in "descending order". Reversing the order of the summation above to put its terms in descending order, we get a series expansion of:

*ar ^{n}*

We can take the common factor of "*a*" out front:

*a*(*r ^{n}*

A basic property of polynomials is that if you divide *x ^{n}* – 1 by

*x ^{n}*

This means that:

If we reverse both subtractions in the fraction above, we will obtain the following equivalent equation:

Applying the above to the geometric summation (by using "*r*" instead of "*x*"), we get:

The above derivation can be extended to give the formula for infinite series, but requires tools from calculus. For now, just note that, for | *r* | < 1, a basic property of exponential functions is that *r ^{n}* must get closer and closer to zero as

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