Discussion
Find the Z-transform of following function:
\(x(n) = a^n u(n) + b^n u(-n - 1)\)
Answer: D
The first term is a right sided sequence and the second is a left sided one, and each has a standard transform.
\[ a^n u(n) \leftrightarrow \frac{z}{z - a},\ |z| \gt |a| \]
\[ b^n u(-n-1) \leftrightarrow -\frac{z}{z - b},\ |z| \lt |b| \]
Adding them gives
\[ X(z) = \frac{z}{z - a} - \frac{z}{z - b} \]
The minus sign comes from the left sided term, so the options with a plus are wrong, and the options without \(z\) in the numerator are the transforms of delayed sequences. The answer is option D.
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