Geometric Progression

For a geometric progression (G.P.), the general term is given by:

For the first G.P.:

First term:

Common ratio:

General term:

For the second G.P.:

First term:

Common ratio:

General term:

Equating both general terms:

Multiplying both sides by :

Since , dividing by 2 gives:

Rewriting as :

Equating exponents:

Thus, the required value of n is: 5.

1. The given geometric progression (G.P.) is:

2, 1, , , , . . .

2. The formula for the th term of a G.P. is:

Where is the first term, is the common ratio, and is the term number.

3. Finding the common ratio:

The common ratio is the ratio of any term to the previous term:

4. Finding the 12th term:

Substitute , , and into the formula:

1. Understanding a Geometric Progression (G.P.):

A sequence is a geometric progression if each term after the first is obtained by multiplying the previous term by a constant ratio. This constant ratio is called the common ratio ( ).

2. Analyzing the Options:

Option (1):

In this sequence, the ratio between consecutive terms is:

The common ratio is 2, so this is a geometric progression.

Option (2):

In this sequence, the ratio between consecutive terms is:

The ratio is not constant (2 and 3), so this is not a geometric progression.

Option (3):

In this sequence, the ratio between consecutive terms is:

The ratio is not constant ( and ), so this is not a geometric progression.

Option (4):

In this sequence, the ratio between consecutive terms is:

The ratio is not constant, so this is not a geometric progression.

3. Conclusion:

The only geometric progression is option (1): .

Correct Answer: (1)