Geometric Progression

In this page geometric progression we are going to see the definition of geometric progression,general form of G.P and the meaning of each term in the formula. Consider the following sequences

(i)   5 ,10,20,40,.................

(ii)  3,6,9,12,15,..................

(iii)  8,16,24,32,..................

In these sequence each term except the first term is obtained by multiplying the proceeding term by a constant. These kind of sequence is known as G.P

Definition

A sequence is said to be a Geometric progression if the ratio of each term,except the first one to its proceeding is always by a constant.This constant ratio is called the common ratio of the G.P is denoted by r.

General form of G.P

a,ar,ar2,ar3,.................

Here a = first term

r = common ratio

to find the common ratio we need to use the formula

r = t2 / t1
here t2 = second terms and t1 = first term

Example 1:

The sum of the first two terms of a G.P is -8 and the sum of the first four terms is -80.Find the first term and the common ratio of the G.P

Solution:

The sum of first two terms = -8

The first four terms are in G.P are

a,ar,ar2,ar3

a + ar = -8

a [ 1 + r ]= -8   ----------- (1)

Also given the sum of first four terms = -80

a + ar + ar2 + ar3 = -80
a [1+r] + ar2 [1+r] = -80
-8 + (-8) r 2 = -80
(-8) r 2 = -80 + 8
r 2 = -72/(-8)
r 2 = 9

r = 3,-3

Substituting r = 3 in the first equation

a[1+3] = 8

a(4) = 8

a = 8/4

a = 2

Substituting r = -3 in the first equation

a[1+(-3)] = 8

a(2) = 8

a = 8/2

a = 4

Therefore the first term = 4, r = -3 and a = 2, r =  3

Related Topics

Quote on Mathematics

“Mathematics, without this we can do nothing in our life. Each and everything around us is math.

Math is not only solving problems and finding solutions and it is also doing many things in our day to day life.  They are:

It subtracts sadness and adds happiness in our life.

It divides sorrow and multiplies forgiveness and love.

Some people would not be able accept that the subject Math is easy to understand. That is because; they are unable to realize how the life is complicated. The problems in the subject Math are easier to solve than the problems in our real life. When we people are able to solve all the problems in the complicated life, why can we not solve the simple math problems?

Many people think that the subject math is always complicated and it exists to make things from simple to complicate. But the real existence of the subject math is to make things from complicate to simple.”

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