2 Arithmetic and geometric progressions

Consider the sequences:

1 , 4 , 7 , 10 , and 3 , 1 , 1 , 3 ,

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 + 2 d , a + 3 d , , a + ( n 1 ) d ,

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 ,

1 , 1 3 , 1 9 , 1 27 ,

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

a , a r , a r 2 , a r 3 ,

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 = 1 3 .

Task!

Find a , d for the arithmetic sequence 3 , 9 , 15 ,

a = 3 , d = 6

Task!

Find a , r for the geometric sequence 8 , 8 7 , 8 49 ,

a = 8 , r = 1 7

Task!

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

4 , 8 , 16 , 32 ,

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