## Geometric Sequences - The Facts

A geometric series is any series where a term is multiplied by a fixed number to get the next term.
2, 6, 18, 54,...
is a geometric sequence. Each term is multiplied by 3 to get the next term.
2, 2, 2, 2,2,...
is a geometric sequence. Each term is multiplied by 1 to get the next term. This sequence is also an arithmetic sequence. Zero is added to each term to get the next term.
The first term of a geometric sequence is labelled
$a$
, and the constant multiplication factor is called the common ratio and labelled
$r$
.
The general geometric sequence is
$a, \: ar, \: ar^2, \: ar^3,...$
(1)
The nth term is
$ar^{n-1}$
.
We can find an expression for the sum of the first
$n$
terms of a geometric series.
$S-n=a+ar+ar^2+...+ ar^{n-2}+ar^{n-1}$

Multiply this expression by
$r$
to give
$rS-n=ar+ar^2+ar^3+...+ ar^{n-1}+ar^n$

Now subtract. On the right ahnd side all except two terms cancel.
$S_n-rS-n=a-ar^n$

Both sides factorise.
$S_n(1-r)=a(1-r^n) \rightarrow S_n = \frac{a(1-r^n)}{1-r}$

The expression is true for
$r \neq 1$
- if
$r=1$
then the sequence is constant and
$S_n=na$
.
If
$r \lt 1$
we can say more. We can sum the complete series. Since
$r \lt 1$
,
$r^n \rightarrow 0$
as
$n \rightarrow \infty$
, then the sum of the infinite series is
$S=\frac{a}{1-r}$
.

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