Sequences and Series
Geometric Sequences
A geometric sequence is a sequence in which the ratio of any term to the previous term is constant. The explicit formula for a geometric sequence is of the form a n = a 1 r-1 , where r is the common ratio. A geometric sequence can be defined recursively by the formulas a 1 = c, a n+1 = ra n , where c is a constant and r is the common ratio.
The sum of a finite geometric sequence (the value of a geometric series) can
be found according to a simple formula. For a geometric sequence
a
n = a
1
r
n-1
, the sum of the first
n
terms is
S
n = a
1(
.
In calculus, the study of infinite geometric series is very involved. Deciding
whether an infinite geometric series is convergent or divergent, and
finding the limits of infinite geometric series are only two of many topics
covered in the study of infinite geometric series. In this text, we'll only use
one formula for the limit of an infinite geometric series. In the case in which
-1 < r < 1
, the limit of the infinite geometric series
a
1
r
n-1 = 
. This limit of the series is the same as the
sum of the infinite geometric sequence
a
n = a
1
r
n-1
.
