## GEOMETRIC PROGRESSION

In this section, we are going to learn about geometric progression.

What is geometric progression ?

A Geometric Progression is a sequence in which each term is obtained by multiplying a fixed non-zero number to the preceding term except the first term.

The fixed number is called common ratio. The common ratio is usually denoted by r.

General form of geometric progression :

The numbers of the form

a, ar, ar2, ...arn-1

is called a Geometric Progression.

The number ‘a’ is called the first term and number ‘r’ is called the common ratio.

To find the common ratio, we use the formula

r  =  a2/a1

General term of the geometric progression :

tn = a r(n-1)

## Geometric Progression Example Problems

Example 1 :

Find out which of the following sequences are geometric sequences . For those geometric sequences, find the common ratio.

(i) 0.12, 0.24, 0.48,.........

Solution :

t1  =  0.12, t2  =   0.24 and t3  =  0.48

 r  =  t2/t1r  =  0.24 / 0.12r  =  24 / 12r  =  2  ----(1) r  =  t3/t2r  =  0.48 / 0.24r  =  48/24r  =  2  ----(2)

Since the common ratios are same, the given sequence is geometric progression.

The required common ratio is 2.

(ii)  0.004, 0.02, 1, ..........

Solution :

t1  =  0.004, t2  =   0.02 and t3  =  1

 r  =  t2/t1r  =  0.02 / 0.004r  =  20 / 4r  =  5  ----(1) r  =  t3/t2r  =  1 / 0.02r  =  100/2r  =  50  ----(2)

Since the common ratios are not same, the given sequence is not a geometric progression.

The required common ratio is 5.

Example 2 :

Find the  10th term and the common ratio of the geometric sequence

1/4, -1/2, 1, -2,............

Solution :

To find the 10th terms of the G.P we use the formula given below.

tn = a r(n-1)

a  =  1/4,  r = (-1/2) / (1/4) ==> -2 and n = 10

t10  =  (1/4) (-2)(10 - 1)

t10  =  (1/4) (-2)9

t10  =  (1/4) (-512)

t10  =  -128

Therefore the 10th and common ratio of the given geometric sequence are -128 and -2 respectively.

After having gone through the stuff given above, we hope that the students would have understood about geometric progression.

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