**Geometric Progression**

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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,ar**^{2},ar^{3},.................
Here a = first term

r = common ratio

to find the common ratio we need to use the formula

r = t

_{2} / t

_{1}
here t

_{2} = second terms and t

_{1} = 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,ar**^{2},ar^{3}
a + ar = -8

a [ 1 + r ]= -8 ----------- (1)

Also given the sum of first four terms = -80

a + ar + ar

^{2} + ar

^{3} = -80

a [1+r] + ar

^{2} [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.”

geometric progression to sequence