### 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 $a=-1$ , $r=\frac{1}{3}$ .

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

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

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

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

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.