## Geometric Sequences - The Facts

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

.