Geometric Progression (G.P.)
Definition
A G.P.
is a sequence whose first term is non-zero and each of whose succeeding term is
r times the preceding term, where r is some fixed non - zero number, known as
the common ratio of the G.P. .
Term and Sum of n Terms:
If a
is the first term and r the common ratio, then G.P. can be written as a, ar,
ar2, . . . the nth term, an, is given by an
= arn-1. The sum Sn of the first n terms of the G.P. is .
If -1 < r <
1, then the sum of the infinite G.P. a + ar + ar2 +........=
Notes:
- If each term of a
G.P. is multiplied (divided) by a fixed non-zero constant, then the resulting
sequence is also a G.P. with same common ratio as that of the given G.P.
- If each term of a
G.P. (with common ratio r) is raised to the power k, then the resulting
sequence is also a G.P. with common ratio rk. .
- If a1, a2,
a3, .......and b1, b2, b3, .......
are two G.P.’s with common ratios r and r¢
respectively then the sequence a1b1 , a2b2,
a3b3.....is also a G.P. with common
ratio r r¢. - If we have to take
three terms in a G.P., it is convenient to take them as a/r, a, ar. In general,
we take
in case we have
to take (2k+1) terms in a G.P. - If we have to take
four terms in a G.P., it is convenient to take them as a/r3,
a/r, ar, ar3 . In general, we take
, in case we have
to take 2k terms in a G.P. - If a1, a2,
. . . . ,an are in G.P. , then a1 an = a2
an-1 = a3 an-2 = . . . . .
- If a1, a2,
a3, ….… is a G.P. ( each aI > 0), then loga1,
loga2, loga3 ….. is an A.P. The converse is also true.
Geometric Means:
- If three terms are
in G.P., then the middle term is called the geometric mean (G.M.) between
the two. So if a, b, c are in G.P. then b =
is the geometric
mean of a and c. - If a1,
a2......an are non-zero positive numbers then their G.M
(G) is given by
G = (a1a2a3......an)1/n.
If G1, G2,…..Gn are n geometric means between
a and b then
a, G1, G2, …., Gn, b will be a G.P. Here b = a
rn + 1 .
&⇒ r = 
&⇒, Gr = a
where Gr
is the rth mean.
Example 1
The
third term of a G.P. is 7. Find the product of first five terms.
Solution:
Let the terms be 
, a , ar, ar2
&⇒ a = 7.
The product = a5
= 75.
Example 2
Show that 
is a
composite number.
Solution:
We have 
= 1090 +1089 + 1088
+ . . . … . + 102 + 10 +1
= 
= 
= m ´ n , where m and n are
natural numbers ( an-1 is divisible by a-1) ,
Hence is a composite
number.
