Four numbers form a geometric progression. If we increase the second number by 4 and decrease the fourth number by 28, we get an arithmetic progression. Determine these four numbers.
We know that twice a term in an arithmetic progression equals the sum of its adjacent terms. Therefore, if the first term of the geometric progression is $a$ and its common ratio is $q$, then according to the problem,

$$
a + a q^{2} = 2(a q + 4)
$$

which, when rearranged, becomes

$$
a(q-1)^{2} = 8
$$

Furthermore,

when rearranged,

$$
a q(q-1)^{2} = 24
$$

Dividing (2) by (1) gives $(q+1)$, resulting in

$$
q = 3
$$

Substituting this value of $q$ into (1) yields

$$
a = 2
$$

Thus, the four numbers we are looking for are: $2, 6, 18, 54$.