Examine Number Properties: Identifying Patterns in 4, 8, 12, 5, 20

To solve this problem, we will investigate whether the sequence of numbers $4, 8, 12, 5, 20$ possesses any identifiable pattern or property, focusing primarily on identifying an arithmetic pattern.

Let's examine the differences between consecutive terms:

• The difference between $8$ and $4$ is $8 - 4 = 4$.
• The difference between $12$ and $8$ is $12 - 8 = 4$.
• The difference between $5$ and $12$ is $5 - 12 = -7$.
• The difference between $20$ and $5$ is $20 - 5 = 15$.

We observe that the differences between consecutive numbers are not consistent. Since the sequence does not exhibit a constant difference, it is not an arithmetic sequence.

We could also consider checking for a geometric pattern, but since immediate calculations show variations in both differences and potential ratios, this seems unnecessary for a basic sequence like this.

None of the properties we considered (arithmetic or geometric) apply. Thus, we conclude there is no identifiable pattern or property consistent across the entire sequence of numbers $4, 8, 12, 5, 20$.

Therefore, the correct answer is: Does not exist.