Mathematics - Functions Question with Solution | TestHub

$f\left(x\right)=a^x$ satisfies the functional relation, so
$\underset{x=1}{\overset{\infty }{\Sigma }}f\left(x\right)=2$
$\Rightarrow a+a^2+a^3\dots \dots \infty \text{terms} =2$
$\Rightarrow \frac{a}{1-a}=2$

$\Rightarrow a=\frac{2}{3}$
$\Rightarrow f\left(x\right)={\left(\frac{2}{3}\right)}^x$
Now, $\frac{f\left(4\right)}{f\left(2\right)}=\frac{{\left(\frac{2}{3}\right)}^4}{{\left(\frac{2}{3}\right)}^2}=\frac{4}{9}$