Question

If $f(x+y)=f\left(x\right)f\left(y\right)$ and $\sum _{x=1}^{\infty }f\left(x\right)=2,x,y\in N,$ where $N$ 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 $\frac{4}{9}$

Let $f\left(x\right)=a^x$
Given $\sum _{x=1}^{\infty }f\left(x\right)=2$
$\Rightarrow a+a^2+\cdots +a^{\infty }=2$
This is the case of Sum of infinite geometric progression
$\Rightarrow \frac{a}{1-a}=2$
$\Rightarrow a=\frac{2}{3}$
$\Rightarrow f\left(x\right)={\left(\frac{2}{3}\right)}^x$
Thus $\frac{f\left(4\right)}{f\left(2\right)}=\frac{{\left(\frac{2}{3}\right)}^4}{{\left(\frac{2}{3}\right)}^2}=\frac{4}{9}$