Sets, Relations & Functions
Functional Equations / Exponential Functions
nta_pyq_2023_jan
Grade 11
Question:
Let $f(x)$ be a function such that $f(x+y) = f(x) \cdot f(y)$ for all $x, y \in \mathbb{N}$. If $f(1) = 3$ and $\displaystyle\sum_{k=1}^{n} f(k) = 3279$, then the value of $n$ is
Step-by-Step Solution
Key Concept: $f(k) = 3^k$; sum is geometric series $\frac{3(3^n-1)}{2} = 3279$.
$f(k)=3^k$. Sum $= 3+3^2+\cdots+3^n = \frac{3(3^n-1)}{2} = 3279 \Rightarrow 3^n = 2187 \Rightarrow n=7$.
Correct Answer: 3