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
11 and 12 Grade Math
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