Understanding the Number Pattern Question

This question asks us to identify the relationship or pattern between the three numbers in the given sets and then find the option set that follows the same relationship. We are given two example sets to help us deduce the rule: (17, 59, 625) and (8, 44, 784).

A crucial constraint is that we must operate on the whole numbers as given, not by breaking them down into individual digits.

Analyzing the Given Example Sets

Let's look closely at the two sets:

1. Set 1: (17, 59, 625)
2. Set 2: (8, 44, 784)

Observe the third number in each set. Often, in such number pattern problems, the third number might be related to the first two through mathematical operations, or it might be a perfect square, cube, or follow some other specific property.

• In Set 1, the third number is 625. We can check if it's a perfect square: $\sqrt{625} = 25$.
• In Set 2, the third number is 784. We can check if it's a perfect square: $\sqrt{784} = 28$.

It appears the third number in both sets is a perfect square. Let's denote the numbers in a set as (a, b, c). So, for Set 1, a=17, b=59, c=625 and $\sqrt{c}=25$. For Set 2, a=8, b=44, c=784 and $\sqrt{c}=28$.

Now, the task is to find a relationship between the first number (a) and the second number (b) that results in the square root of the third number ($\sqrt{c}$).

• For Set 1: We need to get 25 from 17 and 59.
• For Set 2: We need to get 28 from 8 and 44.

Identifying the Pattern

Let's try different combinations of operations on 'a' and 'b' to get $\sqrt{c}$.

• Consider the sum: $a+b$. For Set 1: $17+59=76$. For Set 2: $8+44=52$. How to get 25 from 76 and 28 from 52? No obvious simple pattern like adding/subtracting a constant or multiplying/dividing.
• Consider the difference: $b-a$. For Set 1: $59-17=42$. For Set 2: $44-8=36$. How to get 25 from 42 and 28 from 36?

Let's focus on the results from the difference: (42, 36) and the target values: (25, 28).

• $42 \to 25$: $42 - 17 = 25$. Notice that 17 is the first number 'a' in Set 1.
• $36 \to 28$: $36 - 8 = 28$. Notice that 8 is the first number 'a' in Set 2.

This suggests a pattern! The square root of the third number might be the difference between the second and first number, minus the first number again.

Let's formalize this pattern: $\sqrt{c} = (b - a) - a$. Simplifying this gives us $\sqrt{c} = b - 2a$.

Let's verify this pattern with the given sets:

• Set 1: (17, 59, 625). $a=17, b=59$. Pattern result: $b - 2a = 59 - 2 \times 17 = 59 - 34 = 25$. $\sqrt{625} = 25$. The pattern holds.
• Set 2: (8, 44, 784). $a=8, b=44$. Pattern result: $b - 2a = 44 - 2 \times 8 = 44 - 16 = 28$. $\sqrt{784} = 28$. The pattern holds.

So, the pattern is: The square root of the third number (c) is equal to the second number (b) minus twice the first number (a), i.e., $\sqrt{c} = b - 2a$. Equivalently, $c = (b - 2a)^2$.

Testing the Pattern on the Options

Now, we apply this pattern to each of the given options to find the set that matches.

We will check if $(b - 2a)^2 = c$ for each option (a, b, c).

Option 1: (22, 78, 784)

• $a=22, b=78, c=784$
• Calculate $b - 2a$: $78 - 2 \times 22 = 78 - 44 = 34$
• Calculate $(b - 2a)^2$: $34^2 = 1156$
• Compare to c: $1156 \neq 784$. This option does not follow the pattern.

Option 2: (12, 69, 2025)

• $a=12, b=69, c=2025$
• Calculate $b - 2a$: $69 - 2 \times 12 = 69 - 24 = 45$
• Calculate $(b - 2a)^2$: $45^2 = 2025$
• Compare to c: $2025 = 2025$. This option follows the pattern.

Option 3: (24, 70, 576)

• $a=24, b=70, c=576$
• Calculate $b - 2a$: $70 - 2 \times 24 = 70 - 48 = 22$
• Calculate $(b - 2a)^2$: $22^2 = 484$
• Compare to c: $484 \neq 576$. This option does not follow the pattern. (Note: $\sqrt{576} = 24$, but our pattern produced 22).

Option 4: (18, 74, 1296)

• $a=18, b=74, c=1296$
• Calculate $b - 2a$: $74 - 2 \times 18 = 74 - 36 = 38$
• Calculate $(b - 2a)^2$: $38^2 = 1444$
• Compare to c: $1444 \neq 1296$. This option does not follow the pattern. (Note: $\sqrt{1296} = 36$, but our pattern produced 38).

Based on the analysis, only Option 2 follows the identified pattern.

Conclusion

The pattern observed in the given sets (17, 59, 625) and (8, 44, 784) is that the square root of the third number is equal to the second number minus twice the first number ($\sqrt{c} = b - 2a$). Option (12, 69, 2025) fits this pattern because $69 - 2 \times 12 = 69 - 24 = 45$, and $45^2 = 2025$.

Revision Table: Key Learnings from Number Pattern

Additional Information on Number Analogy Patterns

Number analogy or number pattern questions are common in reasoning tests. They require you to observe, identify, and apply rules governing sets of numbers. These rules can involve various mathematical operations or properties. Here are some common types of patterns you might encounter:

• Arithmetic Progressions: Numbers increase or decrease by a constant difference.
• Geometric Progressions: Numbers are multiplied or divided by a constant ratio.
• Squares and Cubes: Numbers might be squares, cubes, or related to squares/cubes of other numbers in the set.
• Sums and Differences: The third number might be the sum, difference, product, or quotient of the first two, or a result of operations involving these.
• Combinations: The pattern might combine multiple operations, such as $(a+b) \times k = c$ or $(a-b)^2 + k = c$.
• Digit Operations: Although explicitly disallowed in this specific question's constraint, sometimes patterns involve operations on the individual digits of the numbers (e.g., sum of digits, product of digits). Always read question constraints carefully!

Solving these problems effectively involves systematically testing different potential relationships based on the numbers given. Look for simple relationships first (sums, differences, multiples, squares/cubes) and then move to more complex combinations if needed.