Sets, Relations & Functions
Functional Equations / Exponential
nta_pyq_2023_jan
Grade 11
Question:
Suppose $f: \mathbb{R} \to (0,\infty)$ be a differentiable function such that $5f(x+y) = f(x) \cdot f(y)$, $\forall x, y \in \mathbb{R}$. If $f(3) = 320$, then $\displaystyle\sum_{n=0}^{5} f(n)$ is equal to:
Step-by-Step Solution
Key Concept: From functional equation, $f(0)=5$ and $f(x+1)=4f(x)$; sum is geometric with ratio 4.
$f(0)=5$, $f(n)=5\cdot4^n$. $\sum_{n=0}^5 f(n)=5\cdot\frac{4^6-1}{3}=\frac{5\times4095}{3}=6825$.
Correct Answer: 3