Question:
If $f(x+y)=f(x) f(y)$ and $\sum_{x=1}^{\infty} f(x)=2, x, y \in \mathrm{N}$, where $\mathrm{N}$ is the set of all natural numbers, then the value of $\frac{f(4)}{f(2)}$ is :
Correct Option: , 4
Solution:
Let $f(1)=k$, then $f(2)=f(1+1)=k^{2}$
$f(3)=f(2+1)=k^{3}$
$\sum_{x=1}^{\infty} f(x)=2 \Rightarrow k+k^{2}+k^{3}+\ldots \ldots \infty=2$
$\Rightarrow \frac{k}{1-k}=2 \Rightarrow k=\frac{2}{3}$
Now, $\frac{f(4)}{f(2)}=\frac{k^{4}}{k^{2}}=k^{2}=\frac{4}{9}$