Identify the Series Difference

The objective is to find the series that stands out from the others based on its underlying mathematical pattern.

Analyze Series Patterns

We will examine each series to determine if it forms a geometric progression (GP). A geometric progression is a sequence where each term after the first is obtained by multiplying the previous term by a constant non-zero number called the common ratio ($r$).

• Option A: 0.0016, 0.04, 0.2
  • Calculate the ratio between consecutive terms:
  • $\frac{0.04}{0.0016} = 25$
  • $\frac{0.2}{0.04} = 5$
  • The ratios ($25$ and $5$) are not equal. Thus, this is not a geometric progression.
• Option B: 243, 81, 27
  • Calculate the ratio between consecutive terms:
  • $\frac{81}{243} = \frac{1}{3}$
  • $\frac{27}{81} = \frac{1}{3}$
  • The ratios are equal ($\frac{1}{3}$). Thus, this is a geometric progression with $r = \frac{1}{3}$.
• Option C: $\frac{1}{16}, \frac{1}{4}, \frac{1}{2}$
  • Calculate the ratio between consecutive terms:
  • $\frac{1/4}{1/16} = \frac{16}{4} = 4$
  • $\frac{1/2}{1/4} = \frac{4}{2} = 2$
  • The ratios ($4$ and $2$) are not equal. Thus, this is not a geometric progression.
• Option D: 2401, 49, 7
  • Calculate the ratio between consecutive terms:
  • $\frac{49}{2401} = \frac{1}{49}$
  • $\frac{7}{49} = \frac{1}{7}$
  • The ratios ($\frac{1}{49}$ and $\frac{1}{7}$) are not equal. Thus, this is not a geometric progression.

Determine the Different Series

Based on the analysis:

• Options A, C, and D do not exhibit a consistent common ratio, meaning they are not geometric progressions.
• Option B is a geometric progression, as it maintains a constant common ratio ($r = \frac{1}{3}$) between consecutive terms.

Therefore, Option B is the series that is different from the rest because it is the only geometric progression.