Functions
Functional Equations
Premium Question
Grade 12
Question:
$f : \mathbb{N} \to \mathbb{R}$ satisfies $f(x + y) = f(x) + f(y) + 2xy$ for all $x, y \in \mathbb{N}$, and $f(1) = 1$. Then $f(n)$ equals:
(1) $n^2$
(2) $n(n + 1)/2$
(3) $2^n - 1$
(4) $n^2 + n$
Step-by-Step Solution
Key Concept: Use the recurrence $f(n) = f(n - 1) + f(1) + 2(n - 1)$. Compute $f(2), f(3)$ and spot the pattern, then verify by induction or telescoping.
The detailed step-by-step mathematical proof is available inside the Mathbee app workspace.
Correct Answer: (1)