The fourth proportional to $4$, $a$ and $16a$ is $81$. What is the value of $a$?

By definition, if $x$ is the fourth proportional to three numbers $A$, $B$, and $C$, then:
$$\frac{A}{B}=\frac{C}{x}$$

Given $A=4$, $B=a$, $C=16a$, and $x=81$, we substitute these values into the proportion:

$$\frac{4}{a}=\frac{16a}{81}$$

Cross-multiplying to solve for $a$:

$$4\times 81=16a\times a$$

$$324=16a^2$$

$$a^2=\frac{324}{16}$$

$$a^2=\frac{81}{4}$$

Taking the positive square root on both sides:

$$a=\sqrt{\frac{81}{4}}=\frac{9}{2}$$