If $f(x+y)=f(x) \cdot f(y)$ and $\sum_{x=1}^{\infty} f(x)=2, x, y \in N$, where $N$ is the set of al

Question

If $f(x+y)=f(x) \cdot f(y)$ and $\sum_{x=1}^{\infty} f(x)=2, x, y \in N$, where $N$ is the set of all natural numbers, then the value of $\frac{f(4)}{f(2)}$ is

Professor PC Thomas Classes · Answer

Functional equation indicates $f(x) = a^x$. Sum of infinite GP $a/(1-a) = 2 \Rightarrow a = 2/3$. $f(4)/f(2) = a^2 = 4/9$.