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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