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

• Definition of Geometric Progression
• General Form and General Term of a Geometric Progression
• Sum of n terms of a Geometric Progression
• Definition of Geometric Mean
• Position of a term in a Geometric Progression
• Selection of Terms in Geometric Progression
• Sum of an infinite Geometric Progression
• Geometric Progression Formulae
• Properties of Geometric Progression
• Relation between Arithmetic Means and Geometric Means
• Problems on Geometric Progression

11 and 12 Grade Math 

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