In the double-slit experiment, the results observed depend on the specific measurement or detection method used. The experiment can be conducted in such a way that the results appear more "particle-like" or more "wave-like," but it is not possible to observe both the particle result and the wave result for the same individual particle simultaneously.
When the experiment is set up to observe particle-like behavior, such as by using detectors that can determine which slit the particle passes through, the interference pattern disappears. This is because the act of measuring or detecting which path the particle takes "collapses" the wavefunction, forcing the particle to behave more like a classical particle that goes through one slit or the other. In this case, you would see a pattern of two distinct clusters on the screen corresponding to the two slits.
On the other hand, when the experiment is set up to observe wave-like behavior, such as by not measuring which path the particle takes, the interference pattern emerges. This means that the particle goes through both slits simultaneously and interferes with itself, creating an interference pattern on the screen.
Regarding your question about observing every other particle, every third particle, or every fourth particle, the interference pattern would not be visible if you detect or observe individual particles in this manner. The interference pattern arises from the cumulative behavior of a large number of particles. If you detect only specific particles, you would see a random distribution of impacts on the screen, resembling the behavior of classical particles.
In summary, it is not possible to observe both particle-like and wave-like behavior simultaneously for the same individual particle in the double-slit experiment. The observed behavior depends on the specific setup and detection method used, and observing only specific particles would result in patterns that resemble classical particle behavior rather than the interference pattern associated with wave-like behavior.