Using long division method, find the inverse Z-transform of 10z−1
10z
(z−1)(z−2)

To find the inverse Z-transform of the given expression using the long division method, we begin by simplifying the expression and applying the method to determine the inverse. Let's start with the expression:

\[
\frac{10z^{-1}}{(z-1)(z-2)}
\]

### Step 1: Rewrite the expression.
The given expression can be rewritten as:

\[
X(z) = \frac{10z}{(z-1)(z-2)}
\]

### Step 2: Perform long division.
Now, we divide \(10z\) by \((z-1)(z-2)\). We need to express this as a power series, which will give us the inverse Z-transform in terms of a sequence. First, expand \((z-1)(z-2)\):

\[
(z-1)(z-2) = z^2 - 3z + 2
\]

Now, perform long division of \(10z\) by \(z^2 - 3z + 2\).

1. Divide \(10z\) by \(z^2\), which gives us \( \frac{10}{z} \).
2. Multiply \( \frac{10}{z} \) by \(z^2 - 3z + 2\), and subtract this from \(10z\).
   
   \[
   \frac{10}{z} \times (z^2 - 3z + 2) = 10z - 30 + \frac{20}{z}
   \]
   
   Subtract from \(10z\), we get:

   \[
   (10z - 10z) + 30 - \frac{20}{z} = 30 - \frac{20}{z}
   \]

Now, divide \(30 - \frac{20}{z}\) by \(z^2 - 3z + 2\). This process can be continued, but we are now generating a series. Each term generated gives us a term in the Z-transform sequence.

### Step 3: Find the inverse Z-transform.
The inverse Z-transform is the sequence whose Z-transform corresponds to the power series we generate from long division.

In this case, the inverse Z-transform will be a series involving terms like \(10, -30, 20, \dots\). These terms correspond to the coefficients found in the long division process.

### Final Answer:
The inverse Z-transform is the sequence:

\[
x[n] = 10, -30, 20, \dots
\]

Would you like to go through the division process step-by-step with more details?

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**Next Steps:**
1. Would you like to see the complete long division for each step?
2. Do you want to review Z-transform properties?
3. Would you like a recap on how power series relate to Z-transforms?
4. Should we explore a different method of finding the inverse Z-transform?
5. Would you prefer an example with simpler terms?

**Tip:** The long division method converts the Z-transform into a power series where each term can be easily interpreted as part of the sequence corresponding to the inverse Z-transform.

Using long division method, find the inverse Z-transform of 10z−1 / (z−1)(z−2)

Find the inverse Z-transform of the expression 10z / (z-1)(z-2) using the long division method. This detailed solution involves applying polynomial division and power series expansion to derive the sequence corresponding to the inverse Z-transform. Ideal for students in undergraduate engineering or applied mathematics courses.

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