Complete the number analogy: 25 : 37 :: 49 : ?. Identify the number that follows the same pattern as the first pair.

Introduction / Context:
This is a numerical analogy question where the candidate must identify the hidden pattern relating the first pair of numbers and then apply that pattern to the second pair. The pair 25 : 37 is given, and the question asks for the number that should correspond to 49 in the same way. Such questions are common in quantitative aptitude tests and measure a student s ability to detect patterns involving squares, simple operations, or combinations of operations on integers.

Given Data / Assumptions:

• The given pair is 25 and 37.
• The second first term is 49, and the missing number is represented by a question mark.
• The options are 65, 69, 79, and 81.
• We can assume common number patterns such as squares, cubes, and small arithmetic operations are being used.

Concept / Approach:
A good approach is to look for relationships involving squares because both 25 and 49 are perfect squares. We notice that 25 is 5^2. Now, if we take the next integer, which is 6, and square it, we get 6^2 = 36. If we then add 1, we obtain 37. So 37 can be expressed as (5 + 1)^2 + 1. We can test the same pattern on 49. Since 49 is 7^2, the next integer is 8. Squaring 8 gives 8^2 = 64, and adding 1 produces 65. This matches one of the options, suggesting that 49 should pair with 65 under the same rule.

Step-by-Step Solution:
Step 1: Recognise that 25 is 5^2 and 49 is 7^2, which suggests a pattern involving squares. Step 2: For the first pair, take the square root of 25, which is 5. Step 3: Add 1 to this root to get 6, and then square the result: 6^2 = 36. Step 4: Add 1 to this squared value to obtain 36 + 1 = 37, which matches the second number in the first pair. Step 5: Apply the same rule to 49. Its square root is 7. Step 6: Add 1 to the root: 7 + 1 = 8, then square it: 8^2 = 64. Step 7: Finally, add 1: 64 + 1 = 65. This number appears in the options.

Verification / Alternative check:
We can quickly test whether any simpler rules might apply, such as adding a fixed number to 25 to get 37. That would require adding 12, but 49 plus 12 gives 61, which is not among the options. Another possibility is to look at digit patterns, but they do not provide a consistent rule. The square based rule we discovered is elegant and consistent, and it gives the value 65, which exactly matches option 65. No other option satisfies the discovered pattern. Therefore, the analogy 25 : 37 :: 49 : 65 remains stable under verification.

Why Other Options Are Wrong:
69: This number does not arise from the pattern of taking the next integer after the square root and then using the expression n^2 + 1. It is only a random larger number.
79: Similar to 69, this number does not fit the derived rule and seems unrelated to the squares of nearby integers.
81: Although 81 is itself a square, equal to 9^2, the pattern requires (7 + 1)^2 + 1, not an unrelated square. So 81 does not parallel the relation between 25 and 37.

Common Pitfalls:
A typical pitfall is to look only for simple addition or subtraction patterns and ignore higher level structures like squares. Candidates may incorrectly believe that adding a constant to both 25 and 49 will solve the problem. Another common error is to choose 81 simply because it is a square and seems mathematically significant, even though it does not fit the specific rule used for the first pair. The correct strategy is to notice that both 25 and 49 are squares and to explore patterns involving their square roots, which leads naturally to the rule used above.

Final Answer:
Using the pattern (square root of the first number plus one) squared plus one, the number that completes the analogy 25 : 37 :: 49 : ? is 65.