378. Find the condition under which the product of two arbitrary terms of a geometric progression will be a term of the same progression.
Let there be three terms of a geometric progression: $a q^{n}, a q^{p}, a q^{k}$. According to the problem, it should be:

$$
a q^{n} a q^{p}=a q^{k}
$$

from which

$$
a=q^{k-n-p}
$$

Since the numbers $n, p, k$ are integers, the required condition is that the first term of the progression should be some integer power of the common ratio of the progression.