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 \[S=\frac{a}{1-r}\]
.