Analyzing the Number Relation Pattern

The question asks us to identify a set of numbers that shares the same relationship as the given sets: (2, 125, 3) and (1, 64, 3).

We need to find a pattern or rule that connects the three numbers in each set (let's call them A, B, and C), performing operations only on the whole numbers.

Discovering the Pattern in Given Sets

Let's look at the first set: (2, 125, 3).

• Let A = 2, B = 125, C = 3.
• We need to find a relationship between 2, 125, and 3.
• Consider adding the first and third numbers: A + C = 2 + 3 = 5.
• How is this related to the second number, 125? We know that $5^3 = 5 \times 5 \times 5 = 25 \times 5 = 125$.
• This suggests a possible rule: $(A+C)^3 = B$.

Let's test this rule with the second set: (1, 64, 3).

• Let A = 1, B = 64, C = 3.
• Apply the suggested rule: $(A+C)^3$.
• A + C = 1 + 3 = 4.
• $(A+C)^3 = 4^3 = 4 \times 4 \times 4 = 16 \times 4 = 64$.
• This result, 64, matches the second number (B) in the set.

The pattern $(A+C)^3 = B$ holds true for both given sets.

Applying the Pattern to the Options

Now, we will examine each option set to see which one follows the same pattern $(A+C)^3 = B$.

Option 1: (2, 100, 8)

• A = 2, B = 100, C = 8
• Calculate (A + C): 2 + 8 = 10
• Calculate $(A+C)^3$: $10^3 = 10 \times 10 \times 10 = 1000$
• Compare with B: $1000 \neq 100$
• This set does not follow the pattern.

Option 2: (5, 512, 3)

• A = 5, B = 512, C = 3
• Calculate (A + C): 5 + 3 = 8
• Calculate $(A+C)^3$: $8^3 = 8 \times 8 \times 8 = 64 \times 8 = 512$
• Compare with B: $512 = 512$
• This set follows the pattern.

Option 3: (12, 50, 13)

• A = 12, B = 50, C = 13
• Calculate (A + C): 12 + 13 = 25
• Calculate $(A+C)^3$: $25^3 = 25 \times 25 \times 25 = 625 \times 25 = 15625$
• Compare with B: $15625 \neq 50$
• This set does not follow the pattern.

Option 4: (4, 341, 3)

• A = 4, B = 341, C = 3
• Calculate (A + C): 4 + 3 = 7
• Calculate $(A+C)^3$: $7^3 = 7 \times 7 \times 7 = 49 \times 7 = 343$
• Compare with B: $343 \neq 341$
• This set does not follow the pattern.

Only Option 2 follows the identified pattern $(A+C)^3 = B$.

Conclusion

The set (5, 512, 3) is related in the same way as the numbers of the sets (2, 125, 3) and (1, 64, 3), following the rule where the cube of the sum of the first and third numbers equals the second number.

Revision Table: Number Relation

Additional Information: Number Series Patterns

Number relation and series questions often appear in reasoning and quantitative aptitude sections of exams. They test your ability to identify logical patterns between numbers. Common patterns include:

• Arithmetic progression (constant difference)
• Geometric progression (constant ratio)
• Differences of differences (second-order AP)
• Squares or cubes of numbers
• Sum or product of adjacent numbers
• Alternating patterns
• Combinations of the above

To solve these problems, it's helpful to:

• Look for differences or ratios between numbers.
• Check for squares, cubes, or other powers.
• See if there's a relationship between the first/last numbers and the middle one.
• Practice recognizing common number sequences.