Consider the sequences:
In both, any particular term is obtained from the previous term by the addition of a constant value (3 and respectively). Each of these sequences are said to be an arithmetic sequence or arithmetic progression and has general form:
in which are given numbers. In the first example above whereas, in the second example, . The difference between any two successive terms of a given arithmetic sequence gives the value of which is called the common difference .
Two sequences which are not arithmetic sequences are:
In each case a particular term is obtained from the previous term by multiplying by a constant factor (2 and respectively). Each is an example of a geometric sequence or geometric progression with the general form:
where ‘ ’ is the first term and is called the common ratio , being the ratio of two successive terms. In the first geometric sequence above , and in the second geometric sequence , .
Find for the arithmetic sequence
Find for the geometric sequence
Write out the first four terms of the geometric series with , .
The reader should note that many sequences (for example the harmonic sequences) are neither arithmetic nor geometric.