If f(x+y) =

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 :

  1. (1) $\frac{2}{3}$

  2. (2) $\frac{1}{9}$

  3. (3) $\frac{1}{3}$

  4. (4) $\frac{4}{9}$


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, $\frac{f(4)}{f(2)}=\frac{k^{4}}{k^{2}}=k^{2}=\frac{4}{9}$

 

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