A radioactive nucleus \( {}^A_Z X \) undergoes spontaneous decay in the sequence \( {}^A_Z X \rightarrow {}_{Z-1} B \rightarrow {}_{Z-3} C \rightarrow {}_{Z-2} D \), where \( Z \) is the atomic number of element \( X \). The possible decay particles in the sequence are:

Step 1: Analyze the first decay step from \( {}^A_Z X \) to \( {}_{Z-1} B \). The atomic number decreases by 1 (\( Z \rightarrow Z-1 \)), which corresponds to positron emission (\( \beta^+ \) decay).
Step 2: Analyze the second decay step from \( {}_{Z-1} B \) to \( {}_{Z-3} C \). The atomic number decreases by 2 (\( Z-1 \rightarrow Z-3 \)) and the mass number decreases by 4 (since it's a standard radioactive decay chain), which corresponds to an alpha (\( \alpha \)) particle emission.
Step 3: Analyze the third decay step from \( {}_{Z-3} C \) to \( {}_{Z-2} D \). The atomic number increases by 1 (\( Z-3 \rightarrow Z-2 \)), which corresponds to electron emission (\( \beta^- \) decay).
Step 4: Therefore, the sequence of decay particles is \( \beta^+, \alpha, \beta^- \).