Square-plus constant pattern 4 → 19 suggests n^2 + 3; apply the same rule to 7 → ?

Introduction / Context:
Many numerical analogies rely on a simple quadratic relationship. Here, we test for a constant added to a perfect square.

Given Data / Assumptions:

• 4 maps to 19.
• We check a form f(n) = n^2 + k.

Concept / Approach:
Compute 4^2 = 16. To reach 19, add 3. So f(n) = n^2 + 3.

Step-by-Step Solution:
Apply f(n) to 7.7^2 = 49.49 + 3 = 52.

Verification / Alternative check:
Re-check for 4: 16 + 3 = 19, confirming the constant.

Why Other Options Are Wrong:
49: Only n^2, missing the +3.28 and 68: Do not fit the n^2 + 3 form for n = 7.

Common Pitfalls:
Assuming a variable increment or mistaking the constant as +4 or +7.

Final Answer:
52