Functions
Functional Equations
JEE Main 2023
Grade 12
Question:
$f : \mathbb{N} \to \mathbb{N}$ satisfies $f(x + y) = f(x) \cdot f(y)$ for all $x, y \in \mathbb{N}$. If $f(1) = 3$ and $\sum_{k=1}^n f(k) = 3279$, find $n$.
Step-by-Step Solution
Key Concept: The functional equation forces $f(k) = f(1)^k = 3^k$. Sum the resulting geometric series and solve for $n$.
The detailed step-by-step mathematical proof is available inside the Mathbee app workspace.
Correct Answer: 7