In the analogy 7 : 50 :: 9 : ?, find the missing number that continues the same numerical pattern.

Introduction / Context:
This question is an analogy based number problem in the odd man out and series category. Instead of simply choosing a different number, you are asked to complete a pattern defined by the first pair. Recognizing the underlying rule between the numbers is the core challenge.

Given Data / Assumptions:

• You are given the pair 7 : 50 and asked to extend the pattern to 9 : ?.
• The answer must be chosen from the options 37, 82, 17, and 90.
• Assume a consistent mathematical relationship connects the first number in each pair with the second.

Concept / Approach:
The usual technique in such questions is to search for a simple relation like addition, subtraction, multiplication, squaring, or a combination of these. Since 50 is close to 7 squared, we can suspect a pattern involving n^2 plus or minus a constant. Once you find a rule that fits the first pair neatly, you apply it to the second first number and see which option matches.

Step-by-Step Solution:
Step 1: Consider the first pair 7 and 50. Compute 7^2, which is 49. Notice that 49 + 1 equals 50, so 50 can be expressed as 7^2 + 1. Step 2: This suggests a rule of the form second number = first number squared plus 1, written as n^2 + 1. Step 3: Apply this rule to the second first number, which is 9. Compute 9^2, which is 81. Step 4: Add 1 to get 81 + 1 = 82. Step 5: Compare this result with the available options. The value 82 appears among the options, so 82 is the required missing number.

Verification / Alternative check:
You can also test whether any other simple relation fits the pair 7 and 50. For example, 7 * 7 + 1 equals 50, but forms like 7 * 8 or 7 * 6 plus constants do not match as cleanly and would not generalize well to 9. When you apply 9^2 + 1, the resulting 82 fits exactly one of the choices. No other option can be generated by applying the same squared based rule to 9, which confirms that 82 is the correct answer and the pattern is consistent.

Why Other Options Are Wrong:

• 37: This does not equal 9^2 + 1 or follow from any simple extension of the rule used for 7 and 50.
• 17: Very far from 9^2, so it does not satisfy a squared plus constant pattern.
• 90: Also not equal to 9^2 + 1 and lacks a simple relation with the first term that would match the 7 : 50 pair.

Common Pitfalls:
Some students may attempt random arithmetic operations or guess based on closeness in value without properly identifying the rule. Others may overcomplicate the relationship by involving multiple steps when a simple squared pattern already fits perfectly. In analogy questions, always try to find the simplest consistent rule for the first pair before testing your options.

Final Answer:
82