A geometric series is the sum of a geometric sequence. That is,

a + ar + ar_{2 }+ ar_{3 }+ ar_{4} + ...

where r ≠ 0 is the common ratio and a is a scale factor, equal to the sequence's start value.

3 + 6 + 12 + 24 + 48 +... is a geometric serie because each term is obtained by multiplying the preceding number by 2.

To solve exercises using geometric sequences you need the following formulas:

The nth term: a_{n}=a_{1}·r^{n-1}

The sum of the first n terms:

The sum of an infinite geometric serie: only if |r|<1.

where:

a_{1} = the first term of the sequence
r
= common rate n = number of terms a_{n} = nth term
S_{n} = sum of the first n terms
S= sum of an infinite geometric serie

Given , find S_{7}
This is a geometric serie with a common rate r=3.

Determine the sum of the first
7 terms of the geometric sequence where
_{4}= -128 and _{5}= -512