If f(x+y)=f(x)f(y) and _x=1^f(x)=2, x,yN, then the value of f(4)f(2) is:

Question

If $f(x+y)=f(x)f(y)$ and $\displaystyle\sum_{x=1}^{\infty}f(x)=2$, $x,y\in\mathbb{N}$, then the value of $\dfrac{f(4)}{f(2)}$ is:

Accepted Answer

The functional equation forces $f(x)=r^x$ for some $r$. $\displaystyle\sum_{x=1}^{\infty}r^x = \dfrac{r}{1-r}=2 \implies r=\dfrac23.$

$$\frac{f(4)}{f(2)} = r^2 = \frac49.$$

Solutions · 1