# Position of a Term in a Geometric Progression

We will learn how to find the position of a term in a Geometric Progression.

On finding the position of a given term in a given Geometric Progression

We need to use the formula of nth or general term of a Geometric Progression tn = ar$$^{n - 1}$$.

1. Is 6144 a term of the Geometric Progression {3, 6, 12, 24, 48, 96, .............}?

Solution:

The given Geometric Progression is {3, 6, 12, 24, 48, 96, .............}

The first terms of the given Geometric Progression (a) = 3

The common ratio of the given Geometric Progression (r) = $$\frac{6}{3}$$ = 2

Let nth term of the given Geometric Progression is 6144.

Then,

⇒ t$$_{n}$$ = 6144

⇒ a r$$^{n - 1}$$ = 6144

⇒ 3 (2)$$^{n - 1}$$ = 6144

⇒ (2)$$^{n - 1}$$ = 2048

⇒ (2)$$^{n - 1}$$ = 2$$^{11}$$

⇒ n - 1 = 11

⇒ n = 11 + 1

⇒ n = 12

Therefore, 6144 is the 12th term of the given Geometric Progression.

2. Which term of the Geometric Progression 2, 1, ½, ¼, ............. is $$\frac{1}{128}$$?

Solution:

The given Geometric Progression is 2, 1, ½, ¼, .............

The first terms of the given Geometric Progression (a) = 2

The common ratio of the given Geometric Progression (r) = ½

Let nth term of the given Geometric Progression is $$\frac{1}{128}$$.

Then,

t$$_{n}$$ = $$\frac{1}{128}$$

⇒ a r$$^{n - 1}$$ = $$\frac{1}{128}$$

⇒ 2 (½)$$^{n - 1}$$ = $$\frac{1}{128}$$

⇒ (½)$$^{n - 1}$$ = (½)$$^{7}$$

⇒ n - 2 = 7

⇒ n = 7 + 2

⇒ n = 9

Therefore, $$\frac{1}{128}$$ is the 9th term of the given Geometric Progression.

3. Which term of the Geometric Progression 7, 21, 63, 189, 567, ............. is 5103?

Solution:

The given Geometric Progression is 7, 21, 63, 189, 567, .............

The first terms of the given Geometric Progression (a) = 7

The common ratio of the given Geometric Progression (r) = $$\frac{21}{7}$$ = 3

Let nth term of the given Geometric Progression is 5103.

Then,

t$$_{n}$$ = 5103

⇒ a r$$^{n - 1}$$ = 5103

⇒ 7 (3)$$^{n - 1}$$ = 5103

⇒ (3)$$^{n - 1}$$ = 729

⇒ (3)$$^{n - 1}$$ = 3$$^{6}$$

⇒ n - 1 = 6

⇒ n = 6 + 1

⇒ n = 7

Therefore, 5103 is the 7th term of the given Geometric Progression.

Geometric Progression