Understanding Number Set Analogy

The question asks us to find a set of numbers among the given options that follows a similar pattern or relationship as the set (5, 25, 75). To solve this type of problem, we first need to carefully analyze the given set to discover the rule or pattern connecting its elements.

Analyzing the Given Set (5, 25, 75)

Let the given set be represented as (a, b, c), where a = 5, b = 25, and c = 75. We look for mathematical relationships between these numbers.

• Relationship between the first and second number: 25 is the square of 5. So, $b = a^2$.
• Relationship between the second and third number: 75 is 3 times 25. So, $c = b \times 3$.
• Combining these relationships, we can express the third number in terms of the first number: $c = b \times 3 = a^2 \times 3$.

So, the pattern for the given set (5, 25, 75) appears to be:

First term = a

Second term = $a^2$

Third term = $a^2 \times 3$

Checking the Options Against the Pattern

Now, we will test each option to see if it fits the pattern (a, $a^2$, $a^2 \times 3$).

Option 1: (8, 64, 96)

• First term (a) = 8
• Second term based on pattern: $a^2 = 8^2 = 64$. This matches the second number in the option.
• Third term based on pattern: $a^2 \times 3 = 64 \times 3 = 192$.

The set following the pattern with a=8 would be (8, 64, 192). This does not match (8, 64, 96).

Option 2: (4, 16, 48)

• First term (a) = 4
• Second term based on pattern: $a^2 = 4^2 = 16$. This matches the second number in the option.
• Third term based on pattern: $a^2 \times 3 = 16 \times 3 = 48$. This matches the third number in the option.

The set following the pattern with a=4 is (4, 16, 48). This exactly matches Option 2.

Option 3: (8, 32, 64)

• First term (a) = 8
• Second term based on pattern: $a^2 = 8^2 = 64$.

The second number in the option is 32, which does not match 64. So, this option does not follow the pattern.

Option 4: (4, 16, 64)

• First term (a) = 4
• Second term based on pattern: $a^2 = 4^2 = 16$. This matches the second number in the option.
• Third term based on pattern: $a^2 \times 3 = 16 \times 3 = 48$.

The third number in the option is 64, which does not match 48. So, this option does not follow the pattern.

Conclusion

Based on the analysis, Option 2 is the only set that follows the same pattern (a, $a^2$, $a^2 \times 3$) as the given set (5, 25, 75).

Revision Table: Key Pattern Analysis

Understanding number patterns is crucial for solving analogy questions. For the set (a, b, c) similar to (5, 25, 75), we established the relationship b = $a^2$ and c = $b \times 3$, which simplifies to c = $a^2 \times 3$. This specific pattern involving squaring the first term for the second, and then multiplying the second term by a constant (3) for the third, was the key to finding the similar set.

Additional Information: Number Series and Patterns

Number series and pattern questions are common in logical reasoning and quantitative aptitude tests. These questions require identifying the underlying rule that governs a sequence or set of numbers. The patterns can involve:

• Arithmetic progression (constant difference)
• Geometric progression (constant ratio)
• Squaring or cubing numbers
• Adding or subtracting constants
• Multiplying or dividing by constants
• Alternating patterns
• Combinations of the above
• Relationships between terms using their position or value (like in this question, relating terms based on the first term's value).

Practice with various types of number patterns helps in quickly identifying the rule during exams.