.. _section-geometric-sequences:

Geometric Sequences
===================

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

.. math::
   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:

.. math::
   a_n = a r^{n-1}

A geometric sequence follows the recursion relation:

.. math::
   a_n = r a_{n-1}

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

.. math::
   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:

.. math::
   S_\infty = a \sum_{n=1}^{\infty} r^{n-1} = a \sum_{n=0}^{\infty} r^{n} = \frac{a}{1-r}