## HOW TO FIND THE VALUE OF N IN GEOMETRIC PROGRESSION

How to Find the Value of n in Geometric Progression ?

In this section, we will learn how to find the value of n in the geometric progression.

Example 1 :

If the geometric sequences

162, 54, 18,......... and 2/81, 2/27, 2/9,..........

have their nth term equal, find the value of n.

Solution :

Let us find nth term of the given sequences one by one.

162, 54, 18,.........

a  =  162   r  =  54/162 ==> r = 1/3

t =  a r(n-1)

= 1 62 (1/3)(n-1)

=  162 (3-1)(n-1)

=  162 (3-n+1)  ------ (1)

2/81, 2/27, 2/9,..........

a  =  2/81, r  =  (2/27) / (2/81)

r  =  (2/27)  (81/2)

r = 3

tn = a r(n-1)

=  (2/81)  (3)(n-1)  ------ (2)

tn  =  t

162 (3-n+1)  =  (2/81) x 3n-1

[(162  81)/2] (3-n + 1)  =  3n-1

(81  81)(3-n + 1)  =  3n-1

(3⁴ x 3⁴)(3-n + 1)  =  3n-1

(3⁸)(3-n + 1)  =  3n-1

3-n + 9  =  3n-1

-n + 9  =  n - 1

2n  =  10

n  =  5

Therefore the 5th term of the sequence are equal.

## Find the First Three Terms of a Geometric Sequence When Sum of Three Terms is Given

Example 2 :

The sum of three terms of a geometric sequence is 39/10 and their product is 1. Find the common ratio and the terms.

Solution :

Let the first three terms are a/r, a, ar

Sum of three terms  =  39/10

Product of three terms  =  1

(a/r) + a + a r  =  39/10   ------ (1)

(a/r)  a  a r = 1

a3  =  13

a  =  1

By applying the value of a in the first equation, we get

(1/r) + 1 + 1 r  =  39/10

(1 + r + r²)/r  =  39/10

10(1 + r + r²)  =  39 r

10r² + 10r - 39r + 10  =  0

10 r² - 29 r + 10  =  0

(2r - 5)(5r - 2)  =  0

 2r - 5  =  02r  =  5r  =  5/2If a  =  1 and r  =  5/2a/r  =  1/(5/2)a/r  =  2/5a  =  1ar  =  1(5/2)ar  =  5/2 5r - 2  =  05r  =  2r  =  2/5If a  =  1 and r  =  2/5a/r  =  1/(2/5)a/r  =  5/2a  =  1ar  =  1(2/5)ar  =  2/5

Therefore the three terms are

2/5, 1, 5/2 or 5/2, 1, 2/5.

After having gone through the stuff given above, we hope that the students would have understood how to find the value of n in geometric progression.

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