Final answer:
The z transform of x[n]=(1/2)ⁿᵢ[n] + (1/3)ᵢ[n] is X(z) = 1/(1 - (1/2)z⁻¹) + 1/(1 - (1/3)z⁻¹), with the region of convergence |z| > 1/2.
Step-by-step explanation:
The student has asked for the z transform of the sequence x[n]=(1/2)ⁿᵢ[n] + (1/3)ᵢ[n]. To find the transform, we compute the z-transform of each term separately and then add them together.
The z-transform of (1/2)ⁿᵢ[n] is ∑ (1/2)ⁿz⁻ⁿ from n=0 to ∞, which equals to 1/(1 - (1/2)z⁻¹), provided that |z| > |1/2|. The z-transform of (1/3)ᵢ[n] is ∑ (1/3)ⁿz⁻ⁿ from n=0 to ∞, which equals to 1/(1 - (1/3)z⁻¹), provided that |z| > |1/3|.
Combining these results, the overall z-transform of x[n] is:
X(z) = 1/(1 - (1/2)z⁻¹) + 1/(1 - (1/3)z⁻¹), with the region of convergence being |z| > max(|1/2|, |1/3|), i.e. |z| > 1/2.