Understanding Number Analogy Patterns

This question asks us to find a missing number in a sequence based on a specific relationship or pattern observed between pairs of numbers. We are given three pairs, with one number missing in the second pair. The relationship between the first number and the second number in the first pair should be the same as the relationship between the first number and the second number in the third pair. We need to identify this pattern and apply it to the second pair.

The pairs provided are:

• 14 ∶ 197
• 18 ∶ ?
• 16 ∶ 257

Let's analyze the relationship in the first pair: 14 and 197.

We can look for common mathematical operations like squaring, cubing, addition, subtraction, multiplication, or division. Let's consider squaring the first number:

$\text{14}^2 = 14 \times 14 = 196$

The second number in the pair is 197. We can see that $197 = 196 + 1$.

So, the relationship for the first pair seems to be: $(\text{First number})^2 + 1 = \text{Second number}$.

Let's test this potential relationship with the third pair: 16 and 257.

Using the same pattern, we square the first number (16) and add 1:

$\text{16}^2 = 16 \times 16 = 256$

Now, add 1 to the result:

$256 + 1 = 257$

This matches the second number in the third pair (257). The pattern $(\text{First number})^2 + 1 = \text{Second number}$ holds true for both the first and the third pairs.

Now, we can apply this discovered pattern to the second pair to find the missing number. The second pair is 18 ∶ ?.

The first number in this pair is 18. Following the pattern, we need to square 18 and add 1:

$\text{18}^2 = 18 \times 18 = 324$

Now, add 1 to the result:

$324 + 1 = 325$

So, the missing number is 325.

Let's check the given options:

1. 126
2. 526
3. 325
4. 217

Our calculated missing number, 325, matches option 3.

Number Analogy Pattern Summary

The underlying pattern for this number analogy is that the second number in each pair is obtained by squaring the first number and then adding 1.

Revision Table: Analyzing Number Patterns

Additional Information: Types of Number Analogies

Number analogies are common in logical reasoning and quantitative aptitude tests. They require identifying the rule or pattern that relates two numbers in a pair and applying that rule to find a missing number in another pair. Common types of patterns include:

• Arithmetic Operations: Simple addition, subtraction, multiplication, or division.
• Powers and Roots: Squaring, cubing, taking square roots or cube roots.
• Combinations of Operations: A mix of operations, like squaring and adding, multiplying and subtracting, etc. (as seen in this problem).
• Digit Operations: Patterns based on the digits of the number (e.g., sum of digits, product of digits).
• Series Based Patterns: The numbers might follow a sequence like prime numbers, Fibonacci series, etc.

Solving number analogies involves careful observation, testing different mathematical relationships, and verifying the identified pattern across all given pairs before applying it to find the missing term.