Using residues method, find the inverse Z-transform of z2
−3z
(z+ 2)(z−5)

To find the inverse Z-transform of \( X(z) = \frac{z^2 - 3z}{(z + 2)(z - 5)} \) using the residue method, we proceed as follows:

### Step 1: Partial Fraction Decomposition
The given expression is:
\[
X(z) = \frac{z^2 - 3z}{(z + 2)(z - 5)}
\]
We want to express this in the form:
\[
X(z) = \frac{A}{z + 2} + \frac{B}{z - 5}
\]

To find \( A \) and \( B \), multiply both sides by \( (z + 2)(z - 5) \):
\[
z^2 - 3z = A(z - 5) + B(z + 2)
\]
Expanding this:
\[
z^2 - 3z = (A + B)z + (-5A + 2B)
\]

Equating coefficients, we get:
1. \( A + B = 1 \)
2. \( -5A + 2B = -3 \)

Solving these equations:
1. \( A + B = 1 \)
2. \( -5A + 2B = -3 \)

From the first equation, \( B = 1 - A \). Substituting into the second equation:
\[
-5A + 2(1 - A) = -3
\]
\[
-5A + 2 - 2A = -3
\]
\[
-7A + 2 = -3
\]
\[
-7A = -5 \implies A = \frac{5}{7}
\]

Substituting \( A \) back into \( B = 1 - A \):
\[
B = 1 - \frac{5}{7} = \frac{2}{7}
\]

Thus, the partial fraction decomposition is:
\[
X(z) = \frac{5/7}{z + 2} + \frac{2/7}{z - 5}
\]

### Step 2: Finding the Inverse Z-transform
The inverse Z-transform of \( X(z) \) is determined by examining the individual terms.

#### Term 1: \( \frac{5/7}{z + 2} \)
We rewrite this as:
\[
\frac{5/7}{1 - (-2/z)}
\]
The inverse Z-transform of \( \frac{1}{1 - a/z} \) is \( a^n u[n] \). Thus, the inverse Z-transform of \( \frac{5/7}{z + 2} \) corresponds to \( \left(\frac{-2}{1}\right)^n u[n] \), giving:
\[
\frac{5}{7}(-2)^n u[n]
\]

#### Term 2: \( \frac{2/7}{z - 5} \)
We rewrite this as:
\[
\frac{2/7}{1 - (5/z)}
\]
The inverse Z-transform of this is \( \left(\frac{5}{1}\right)^n u[n] \), giving:
\[
\frac{2}{7}(5)^n u[n]
\]

### Combining both results
The overall inverse Z-transform is:
\[
x[n] = \frac{5}{7}(-2)^n u[n] + \frac{2}{7}(5)^n u[n]
\]

### Final Answer
\[
x[n] = \frac{5}{7}(-2)^n u[n] + \frac{2}{7}(5)^n u[n]
\]

Would you like me to elaborate on any of these steps or verify the calculations?

### Relative Questions
1. What is the significance of using the residue method in inverse Z-transforms?
2. How do we handle partial fraction decomposition for more complex expressions?
3. What are common properties of Z-transforms that help simplify inverse calculations?
4. How does the region of convergence affect the inverse Z-transform?
5. Can the residue method be applied to Z-transforms with higher-order poles?

### Tip
Always check the region of convergence when applying the inverse Z-transform, as it ensures the solution corresponds to the correct sequence behavior (causal vs. non-causal).

Using residues method, find the inverse Z-transform of z^2 − 3z / ((z + 2)(z − 5))

Discover how to compute the inverse Z-transform of z^2 − 3z / ((z + 2)(z − 5)) using the residue method. This solution covers partial fraction decomposition and inverse Z-transform concepts, suitable for undergraduate students in engineering or mathematics.

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