2 Arithmetic and geometric progressions

Consider the sequences:

$1,4,7,10,\dots$ and $3,1,-1,-3,\dots$

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:

$a,a+d,a+2d,a+3d,\dots ,a+\left(n-1\right)d,\dots$

in which $a,d$ are given numbers. In the first example above $a=1,d=3$ whereas, in the second example, $a=3,d=-2$ . 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:

$1,2,4,8,\dots$

$-1,-\frac{1}{3},-\frac{1}{9},-\frac{1}{27},\dots$

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:

$a,ar,a{r}^2,a{r}^3,\dots$

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 $3,9,15,\dots$

Answer

Find $a,r$ for the geometric sequence $8,\frac{8}{7},\frac{8}{49},\dots$

Answer

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

Answer

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