Analyzing the Analogous Pair and Finding the Missing Term

The problem presents an analogous pair with four terms: 2: 12, 5 : 122 : : 4 : 252, 7 : _______. We need to find the missing term that completes the analogy for the last pair (7 : ?).

An analogous pair implies that the relationship between the first two terms (2 and 12, and 5 and 122) is similar, and the relationship between the third and fourth terms (4 and 252, and 7 and ?) is also similar. Often, the pairs within the first set share a rule, and the pairs within the second set share another rule. Let's analyze the given pairs to find the underlying pattern.

Identifying the Number Pattern

Let's examine how the second number is derived from the first number in each given pair:

1. Pair 1: 2 → 12
2. Pair 2: 5 → 122
3. Pair 3: 4 → 252
4. Pair 4: 7 → ?

We can try different mathematical operations or relationships involving powers of the first number to obtain the second number. Let's consider squares (\(n^2\)), cubes (\(n^3\)), or even higher powers (\(n^4\)).

Analyzing the Relationship for Each Pair

• For 2 → 12:
  • \(2^2 = 4\). \(4 + 8 = 12\).
  • \(2^3 = 8\). \(8 + 4 = 12\). (Pattern: \(n^3 + 2n^2\) doesn't work, \(n^3+n^2\) works just for n=2)
  • \(2^4 = 16\). \(16 - 4 = 12\). (Pattern: \(n^4 - 4\))
• For 5 → 122:
  • \(5^2 = 25\). \(25 + 97 = 122\).
  • \(5^3 = 125\). \(125 - 3 = 122\). (Pattern: \(n^3 - 3\))
• For 4 → 252:
  • \(4^2 = 16\). \(16 + 236 = 252\).
  • \(4^3 = 64\). \(64 + 188 = 252\).
  • \(4^4 = 256\). \(256 - 4 = 252\). (Pattern: \(n^4 - 4\))
• For 7 → ?:
  • \(7^2 = 49\).
  • \(7^3 = 343\).
  • \(7^4 = 2401\).

Discovering the Pattern Based on Number Type (Even/Odd)

Let's observe the patterns that seemed to work for individual pairs:

• For 2 (even): \(2^4 - 4 = 12\)
• For 5 (odd): \(5^3 - 3 = 122\)
• For 4 (even): \(4^4 - 4 = 252\)

It appears there might be two different rules in play, one for even numbers and one for odd numbers.

• Hypothesis 1: For even numbers (\(n\)), the rule is \(n^4 - 4\).
• Hypothesis 2: For odd numbers (\(n\)), the rule is \(n^3 - 3\).

Verifying the Hypotheses

Let's check if these rules hold for the given pairs:

The rules fit all the given pairs perfectly. Now, we apply the rule for odd numbers to find the missing term for \(n=7\).

Calculating the Missing Term

The missing term corresponds to the number 7, which is an odd number. According to our discovered pattern, the rule for odd numbers is \(n^3 - 3\).

For \(n=7\):

Missing Term \(= 7^3 - 3\)

Missing Term \(= 343 - 3\)

Missing Term \(= 340\)

Thus, the missing term in the analogous pair is 340.

Conclusion

The analogy follows two patterns based on whether the input number is even or odd. For even numbers, the pattern is \(n^4 - 4\). For odd numbers, the pattern is \(n^3 - 3\). Applying the odd number pattern to 7 gives \(7^3 - 3 = 340\).

Revision Table: Analogous Pair Pattern

Additional Information: Solving Analogies

Analogies in logical reasoning tests require identifying the relationship between a given pair of terms and applying that same relationship (or a parallel relationship, as seen in this example) to a different term to find a missing term. The relationship can be based on various principles:

• Mathematical operations (addition, subtraction, multiplication, division, powers, roots)
• Sequential patterns
• Letter sequences (position in alphabet)
• General knowledge relationships (e.g., country-capital, animal-sound)
• Odd/Even number properties
• Prime/Composite number properties

It's important to systematically test different possible relationships and check for consistency across the given pairs in the analogy.