In the following number analogy, 2 is related to 10 in a particular way. Using the same pattern, which number should replace the question mark in 2 : 10 :: 3 : ?

Introduction / Context:
This analogy involves a numerical pattern based on a small algebraic expression. The number 2 is related to 10 through a specific formula, and we must apply that same formula to 3. Problems of this sort test your ability to detect patterns involving both multiplication and powers, and then generalise those patterns to new inputs.

Given Data / Assumptions:
First pair: 2 : 10. Second pair: 3 : ?. All numbers are integers. The same mathematical rule must be used for both pairs.

Concept / Approach:
We start by experimenting with 2 to see how we can obtain 10 using a reasonable combination of its square or cube and a multiplier. A neat pattern that often appears in such questions is n multiplied by (n^2 + 1). If this works cleanly for 2 and produces 10, we then apply n * (n^2 + 1) to 3 and see whether the resulting value is present among the options. The aim is to find a simple polynomial expression rather than a complicated or irregular rule.

Step-by-Step Solution:
Step 1: Test the expression n * (n^2 + 1) for n = 2. Compute 2^2 = 4. Then n^2 + 1 = 4 + 1 = 5. Now multiply by n: 2 * 5 = 10. This exactly matches the first pair, so the rule works for 2. Step 2: Apply the same rule to n = 3. Compute 3^2 = 9. Then n^2 + 1 = 9 + 1 = 10. Now multiply by n: 3 * 10 = 30. So the required number is 30. Step 3: Check whether 30 appears among the options. 30 is present, so it is the correct answer.

Verification / Alternative check:
We can rephrase the pattern as output = n * (n^2 + 1). For n = 2, the formula gives 2 * (4 + 1) = 2 * 5 = 10. For n = 3, the formula gives 3 * (9 + 1) = 3 * 10 = 30. No other simple integer n substituted into this expression will equal 10 for the first pair, which confirms that the rule fits tightly and is not arbitrary. Because 30 is the only option that can be expressed in the form 3 * (3^2 + 1), the analogy is satisfied uniquely by that value.

Why Other Options Are Wrong:
25 is 5^2 and does not match 3 * (3^2 + 1). 17 could be approximated by 3^2 + 8, but that does not connect in a clean, parallel way with the transformation from 2 to 10. 35 and 19 also do not arise from multiplying 3 by (3^2 + 1), and they would require different, unrelated rules.

Common Pitfalls:
Many students will first try very simple rules like multiplying by a constant or adding a fixed number and may give up when those do not work. It is helpful to remember that exam setters often use expressions involving n^2 or n^3 combined with n. When you suspect such a pattern, compute n^2 and n^3 and try small variations like n * (n^2 + 1). Always ensure that the same expression works for every pair given in the question before finalising your answer.

Final Answer:
The number that correctly completes the analogy is 30.