To solve this problem, we need to analyze the decay sequence of the radioactive nucleus $^{A}_{Z}X. \\$The sequence is: $^{A}_{Z}X \rightarrow _{Z-1}B \rightarrow _{Z-3}C \rightarrow _{Z-2}D. \\$Let's break down each step of the decay:$\\$1. $^{A}_{Z}X \rightarrow _{Z-1}B: \\$• In this step, the atomic number decreases by 1, which indicates a $\beta^+$ decay.$\\$• In $\beta^+$ decay, a proton is converted into a neutron, emitting a positron ($e^+$) and a neutrino.$\\$2. $_{Z-1}B \rightarrow _{Z-3}C: \\$• Here, the atomic number decreases by 2, which is characteristic of $\alpha$ decay.$\\$• In $\alpha$ decay, an $\alpha$ particle ($^{4}_{2}$He$$) is emitted, reducing the atomic number by 2 and the mass number by 4.$\\$3. $_{Z-3}C \rightarrow _{Z-2}D: \\$• In this step, the atomic number increases by 1, indicating a $\beta^-$ decay.$\\$• In $\beta^-$ decay, a neutron is converted into a proton, emitting an electron ($e^-$) and an antineutrino.$\\$Therefore, the sequence of decay particles is: $\beta^+, \alpha, \beta^-. \\$This corresponds to Option 3.