6.5 GEOMETRIC PROGRESSIONS
Consider the lists
(i) 30, 90, 270, 810 ..... (ii)
111 1
, , , .....
4 16 64 256
(iii) 30, 24, 19.2, 15.36, 12.288
Can we write the next term in each of the lists above?
In (i), each term is obtained by multiplying the preceeding term by
3.
In (ii), each term is obtained by multiplying the preceeding term by
1
4
.
In (iii), each term is obtained by multiplying the preceeding term by 0.8.
In all the lists given above, we see that successive terms are obtained by multiplying the
preceeding term by a fixed number. Such a list of numbers is said to form Geometric Progression
(GP).
This fixed number is called the common ration ‘r’ of GP. So in the above example (i), (ii),
(iii) the common ratios are 3,
1
4
, 0.8 respectively.
Let us denote the first term of a GP by a and common ratio r. To get the second term
according to the rule of Geometric Progression, we have to multiply the first term by the common
ratio r, where a ¹ 0, r ¹ 0 and r ¹ 1
\ The second term = ar
Third term = ar. r = ar
2
\ a, ar, ar
2
..... is called the general form of a GP.
In the above GP the ratio between any term (except first term) and its preceding term is ‘r’
i.e.,
2
..........
ar ar
r
a ar
== =
If we denote the first term of GP by a1
, second term by a2
..... n
th term by an
then 2 3
12 1
......
-
== = =
n
n
aa a
r
aa a
\ A list of numbers a1
, a2
, a3
.... an
... is called a geometric progression (GP), if each
term is non zero and
1
n
n
a
r
a -
= (r ¹ 1)
where n is a natural number and n > 2.