Solution. From the inequality between the arithmetic and geometric means, we have $\frac{4+a_{i}}{5} \geq \sqrt[5]{a_{i}}$, i.e., $1+a_{i} \geq 5 \sqrt[5]{a_{i}}>0$, for $i=1,2, \ldots, n$. If we multiply these $n$ inequalities, we get

$$\left(4+a_{1}\right)\left(4+a_{2}\right) \ldots\left(4+a_{n}\right) \geq 5 \sqrt[5]{a_{1} a_{2} \ldots a_{n}}=5^{n}$$

where the equality holds if and only if $a_{i}=1$, for $i=1,2, \ldots, n$.