6.1. Geometric Sequences

Most of this is from Wikipedia. A geometric sequence is of the form:

\[a, ar, ar^2, ar^3, \cdots\]

where \(r\not=0\) is the common ratio and \(a\) is the scale factor. If we start numbering the values in the sequence at one, the n-th term is given by:

\[a_n = a r^{n-1}\]

A geometric sequence follows the recursion relation:

\[a_n = r a_{n-1}\]

The sequence becomes a series in case we sum all terms in the sequence:

\[S_N = \sum_{n=1}^{N} a_n = a \sum_{n=1}^{N} r^{n-1} = \frac{a(1-r^N)}{1-r}\]

For the limit \(N\rightarrow\infty\) in case \(|r|<1\) we have:

\[S_\infty = a \sum_{n=1}^{\infty} r^{n-1} = a \sum_{n=0}^{\infty} r^{n} = \frac{a}{1-r}\]