Question

If $f\left(x+y\right)=f\left(x\right)·\left(y\right)$ and $\sum _{x=1}^{\infty }f\left(x\right)=2$. $x,y\in \mathrm{\mathbb{N}}$, where $\mathrm{\mathbb{N}}$ is the set of all natural numbers, then the value of $\frac{f\left(4\right)}{f\left(2\right)}$ is

• A. $\frac{2}{3}$
• B. $\frac{1}{9}$
• C. $\frac{1}{3}$
• D. $\frac{4}{9}$

Answer 1

The correct option is D

$\frac{4}{9}$

Explanation for the correct option:

It is given that $f\left(x+y\right)=f\left(x\right)·\left(y\right)$.

The given equation, $\sum _{x=1}^{\infty }f\left(x\right)=2$.

$$\begin{gathered} \Rightarrow f\left(1\right)+f\left(2\right)+f\left(3\right)+.......\infty =2 \\ \Rightarrow f\left(1\right)+f\left(1+1\right)+f\left(1+2\right)+.......\infty =2 \\ \Rightarrow f\left(1\right)+f\left(1\right)·f\left(1\right)+f\left(1\right)·f\left(2\right)+.......\infty =2\left[\because f\left(x+y\right)=f\left(x\right)·\left(y\right)\right] \\ \Rightarrow f\left(1\right)+f\left(1\right)\left[f\left(1\right)+f\left(2\right)+.......\infty \right]=2 \\ \Rightarrow f\left(1\right)+f\left(1\right)\times 2=2\left[\because f\left(1\right)+f\left(2\right)+f\left(3\right)+.......\infty =2\right] \\ \Rightarrow 3f\left(1\right)=2 \\ \Rightarrow f\left(1\right)=\frac{2}{3} \end{gathered}$$

Thus, $\frac{f\left(4\right)}{f\left(2\right)}=\frac{f\left(2+2\right)}{f\left(2\right)}$

$$\begin{gathered} \Rightarrow \frac{f\left(4\right)}{f\left(2\right)}=\frac{f\left(2\right)·f\left(2\right)}{f\left(2\right)}\left[\because f\left(x+y\right)=f\left(x\right)·\left(y\right)\right] \\ \Rightarrow \frac{f\left(4\right)}{f\left(2\right)}=f\left(2\right) \\ \Rightarrow \frac{f\left(4\right)}{f\left(2\right)}=f\left(1+1\right) \\ \Rightarrow \frac{f\left(4\right)}{f\left(2\right)}=f\left(1\right)·f\left(1\right) \\ \Rightarrow \frac{f\left(4\right)}{f\left(2\right)}=\frac{2}{3}·\frac{2}{3} \\ \Rightarrow \frac{f\left(4\right)}{f\left(2\right)}=\frac{4}{9} \end{gathered}$$

Therefore, the value of $\frac{f\left(4\right)}{f\left(2\right)}$ is $\frac{4}{9}$.

Hence, (D) is the correct option.