Understanding Number Pattern Reasoning

This question asks us to identify the relationship between numbers in a given set and find another set from the options that follows the same relationship. We are given two example sets: (17, 34, 68) and (24, 48, 96). We need to analyze these sets to determine the rule connecting the numbers.

Analyzing the Given Number Sets

Let's look at the first set: (17, 34, 68).

• The first number is 17.
• The second number is 34. We can see that $34 = 17 \times 2$.
• The third number is 68. We can see that $68 = 34 \times 2$.

So, in this set, the second number is twice the first, and the third number is twice the second.

Now let's look at the second set: (24, 48, 96).

• The first number is 24.
• The second number is 48. We can see that $48 = 24 \times 2$.
• The third number is 96. We can see that $96 = 48 \times 2$.

This set follows the same pattern: the second number is twice the first, and the third number is twice the second.

Identifying the Number Relationship Pattern

From the analysis of both sets, the pattern is consistent:

If the numbers in the set are represented as $(a, b, c)$, the relationship is:

• $b = a \times 2$
• $c = b \times 2$

This also means $c = (a \times 2) \times 2 = a \times 4$.

So, the pattern is $(a, 2a, 4a)$.

Checking Options Against the Pattern

Now we will examine each option to see which one follows the $(a, 2a, 4a)$ pattern.

Option 1: (31, 62, 106)

• First number (a) = 31
• Is the second number $31 \times 2$? $31 \times 2 = 62$. Yes, the second number is 62.
• Is the third number $62 \times 2$? $62 \times 2 = 124$. The third number in the option is 106.

Since $106 \neq 124$, this option does not follow the pattern.

Option 2: (26, 52, 156)

• First number (a) = 26
• Is the second number $26 \times 2$? $26 \times 2 = 52$. Yes, the second number is 52.
• Is the third number $52 \times 2$? $52 \times 2 = 104$. The third number in the option is 156.

Since $156 \neq 104$, this option does not follow the pattern.

Option 3: (29, 58, 116)

• First number (a) = 29
• Is the second number $29 \times 2$? $29 \times 2 = 58$. Yes, the second number is 58.
• Is the third number $58 \times 2$? $58 \times 2 = 116$. The third number in the option is 116.

Since $116 = 116$, this option follows the pattern $(29, 29 \times 2, 58 \times 2)$, which is $(29, 58, 116)$. This matches the $(a, 2a, 4a)$ relationship.

Option 4: (14, 42, 84)

• First number (a) = 14
• Is the second number $14 \times 2$? $14 \times 2 = 28$. The second number in the option is 42.

Since $42 \neq 28$, this option does not follow the pattern. Although $84 = 42 \times 2$, the first part of the pattern $(a, 2a)$ is not met.

Conclusion

Only Option 3, the set (29, 58, 116), demonstrates the same mathematical relationship $(a, 2a, 4a)$ as the given sets (17, 34, 68) and (24, 48, 96). The numbers in this set are 29, $29 \times 2 = 58$, and $58 \times 2 = 116$ (or $29 \times 4 = 116$).

Revision Table: Number Pattern Analysis

Additional Information on Number Series and Reasoning

Number pattern reasoning questions are common in competitive exams. They test your ability to identify logical or mathematical rules governing a sequence or set of numbers. Common patterns include arithmetic progression, geometric progression, squares, cubes, prime numbers, and combinations of operations like addition, subtraction, multiplication, and division.

• Arithmetic Progression: Each term after the first is obtained by adding a constant difference. Example: (3, 6, 9) - add 3.
• Geometric Progression: Each term after the first is obtained by multiplying by a constant ratio. Example: (3, 6, 12) - multiply by 2. This question uses a pattern related to geometric progression.
• Difference/Ratio Analysis: Finding the differences or ratios between consecutive numbers is a primary method to uncover the underlying pattern.
• Combination Series: Patterns might involve alternating operations or a rule that depends on the position of the number in the sequence.

Practicing various types of number series and set-based reasoning problems helps in quickly recognizing patterns during exams.