Question

Which of the following is a geometric progression?

• A. \(\frac 12, \frac 14, \frac 18,.....\)
• B. \(-2, -4, -12,.....\)
• C. \(3,4,6,12,.....\)
• D. \(x,1,x^2,…..\)

Answer 1

The Correct Option is A

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 (\(r\)).

2. Analyzing the Options:

Option (1): \( 1, 2, 4, 8, 16, \dots \)

In this sequence, the ratio between consecutive terms is:

\( \frac{2}{1} = 2, \quad \frac{4}{2} = 2, \quad \frac{8}{4} = 2, \quad \frac{16}{8} = 2 \)

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

Option (2): \( -2, -4, -12, \dots \)

In this sequence, the ratio between consecutive terms is:

\( \frac{-4}{-2} = 2, \quad \frac{-12}{-4} = 3 \)

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

Option (3): \( 3, 4, 6, 12, \dots \)

In this sequence, the ratio between consecutive terms is:

\( \frac{4}{3} = \frac{4}{3}, \quad \frac{6}{4} = \frac{3}{2} \)

The ratio is not constant (\(\frac{4}{3}\) and \(\frac{3}{2}\)), so this is not a geometric progression.

Option (4): \( x, 1, x^2, \dots \)

In this sequence, the ratio between consecutive terms is:

\( \frac{1}{x} = \frac{1}{x}, \quad \frac{x^2}{1} = x^2 \)

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

3. Conclusion:

The only geometric progression is option (1): \( 1, 2, 4, 8, 16, \dots \).

Correct Answer: (1) \( 1, 2, 4, 8, 16, \dots \)