### 2 Arithmetic and geometric progressions

Consider the sequences:

$\phantom{\rule{2em}{0ex}}1,\phantom{\rule{1em}{0ex}}4,\phantom{\rule{1em}{0ex}}7,\phantom{\rule{1em}{0ex}}10,\phantom{\rule{1em}{0ex}}\dots $ and $3,\phantom{\rule{1em}{0ex}}1,\phantom{\rule{1em}{0ex}}-1,\phantom{\rule{1em}{0ex}}-3,\phantom{\rule{1em}{0ex}}\dots $

In both, any particular term is obtained from the previous term by the
**
addition
**
of a constant value (3 and
$-2$
respectively). Each of these sequences are said to be an
**
arithmetic sequence
**
or
**
arithmetic progression
**
and has general form:

$\phantom{\rule{2em}{0ex}}a,\phantom{\rule{1em}{0ex}}a+d,\phantom{\rule{1em}{0ex}}a+2d,\phantom{\rule{1em}{0ex}}a+3d,\phantom{\rule{1em}{0ex}}\dots ,\phantom{\rule{1em}{0ex}}a+\left(n-1\right)d,\phantom{\rule{1em}{0ex}}\dots $

in which
$a,d$
are given numbers. In the first example above
$a=1,\phantom{\rule{1em}{0ex}}d=3$
whereas, in the second example,
$a=3,\phantom{\rule{1em}{0ex}}d=-2$
. The difference between any two successive terms of a given arithmetic sequence gives the value of
$d$
which is called the
**
common difference
**
.

Two sequences which are
**
not
**
arithmetic sequences are:

$\phantom{\rule{2em}{0ex}}1,\phantom{\rule{1em}{0ex}}2,\phantom{\rule{1em}{0ex}}4,\phantom{\rule{1em}{0ex}}8,\phantom{\rule{1em}{0ex}}\dots $

$\phantom{\rule{2em}{0ex}}-1,\phantom{\rule{1em}{0ex}}-\frac{1}{3},\phantom{\rule{1em}{0ex}}-\frac{1}{9},\phantom{\rule{1em}{0ex}}-\frac{1}{27},\phantom{\rule{1em}{0ex}}\dots $

In each case a particular term is obtained from the previous term by
**
multiplying
**
by a constant factor (2 and
$\frac{1}{3}$
respectively). Each is an example of a
**
geometric sequence
**
or
**
geometric progression
**
with the general form:

$\phantom{\rule{2em}{0ex}}a,\phantom{\rule{1em}{0ex}}ar,\phantom{\rule{1em}{0ex}}a{r}^{2},\phantom{\rule{1em}{0ex}}a{r}^{3},\phantom{\rule{1em}{0ex}}\dots $

where ‘
$a$
’ is the first term and
$r$
is called the
**
common ratio
**
, being the ratio of two successive terms. In the first geometric sequence above
$a=1$
,
$r=2$
and in the second geometric sequence
$a=-1$
,
$r=\frac{1}{3}$
.

##### Task!

Find $a,d$ for the arithmetic sequence $3,\phantom{\rule{1em}{0ex}}9,\phantom{\rule{1em}{0ex}}15,\dots $

$a=3,\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}d=6$

##### Task!

Find $a,r$ for the geometric sequence $8,\phantom{\rule{1em}{0ex}}\frac{8}{7},\phantom{\rule{1em}{0ex}}\frac{8}{49},\phantom{\rule{1em}{0ex}}\dots $

$a=8,\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}r=\frac{1}{7}$

##### Task!

Write out the first four terms of the geometric series with $a=4$ , $r=-2$ .

$4,-8,16,-32,\dots $

The reader should note that many sequences (for example the harmonic sequences) are neither arithmetic nor geometric.