In this number analogy, 4 is related to 17 by a specific operation; using the same rule, determine the number related to 7 in the pair: 4 : 17 :: 7 : ?

Introduction / Context:
This question is from the number analogy section of quantitative aptitude. We are given one pair of numbers, 4 and 17, and must figure out how the second number is obtained from the first. Once we determine that rule, we apply it to 7 to find the missing term. Such questions improve your ability to see patterns in arithmetic operations like squaring, adding and subtracting.

Given Data / Assumptions:

• The first number in the given pair is 4 and the second number is 17.
• In the target pair, the first number is 7 and the second number is unknown.
• The same mathematical operation or pattern must connect 4 to 17 and 7 to the missing number.
• We usually look for simple arithmetic rules such as addition, multiplication or squaring.

Concept / Approach:
Many standard number analogies use squares or cubes combined with a small addition or subtraction. We should therefore test whether 17 can be expressed using 4 in such a way. Observing that 4 squared is 16, we notice that 17 is exactly 4 squared plus 1. Once we verify that this relation is clean and simple, we apply the same pattern to 7.

Step-by-Step Solution:
Step 1: Compute the square of 4: 4^2 = 16.Step 2: Compare 16 with the given second number 17. We see that 17 = 16 + 1 = 4^2 + 1.Step 3: Infer the rule from this: for a given number n, its pair is n^2 + 1.Step 4: Apply this rule to 7. Compute 7^2 = 49.Step 5: Add 1 to this square: 49 + 1 = 50.Step 6: The number related to 7 should therefore be 50.

Verification / Alternative check:
To verify, we can check if any other simpler rule fits both 4 and 17, such as 4 * 4 + 1 or 4 * (4 + 1). We have already seen that 4 * 4 + 1 is the same as 4^2 + 1, which works neatly. None of the options besides 50 are of the form 7^2 plus or minus a small fixed number that would also match 4^2 plus the same adjustment. Hence our rule remains consistent and unique.

Why Other Options Are Wrong:
Option A, 49, is just 7^2 and ignores the addition of 1 that we observed in the first pair. Option C, 51, and Option D, 52, would require rules like 7^2 + 2 or 7^2 + 3, but those do not match 4^2 + 1 for the first pair. Since any valid pattern must work for both pairs, these options are inconsistent and therefore incorrect.

Common Pitfalls:
Students often stop at 7^2 = 49 and choose it hastily without examining the exact way 17 relates to 4. Another pitfall is to look for complicated operations when a simple square plus one rule is enough. Always start by testing basic operations and confirm that the rule works for every given pair in the analogy.

Final Answer:
Using the same pattern, the number related to 7 is 50, so option B is correct.