---
title: "Inverse z of (z+2)/((z-2)(z-3))"
subject: "Applied Mathematics"
chapter: 2
key: "semester_4/applied_mathematics:inverse-z-z-plus-2-over-z-2-z-3"
source: https://www.bctengineer.com/q/semester_4/applied_mathematics/inverse-z-z-plus-2-over-z-2-z-3
site: BCT Engineering
---

# Inverse z of (z+2)/((z-2)(z-3))

Applied Mathematics · Chapter 2: The Z-Transform

- Asked in: 2070 Magh, 5 marks
- Video: [Inverse Z-Transform of z/((z−2)(z−3)) | VTU Exam](https://www.youtube.com/watch?v=Bta7Bd_uIxQ) — Mathematics Tutor
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/applied_mathematics/inverse-z-z-plus-2-over-z-2-z-3
- Whole chapter: https://www.bctengineer.com/md/semester_4/applied_mathematics/chapter-2

## Question

Obtain inverse Z-transform of (z + 2)/((z - 2)(z - 3)).

## Short answer

Use partial fractions of X(z)/z: X(z)=1/3 -2z/(z-2)+(5/3)z/(z-3). Inverse gives x(k)=1/3 delta(k)-2·2^k+(5/3)3^k; for k≥1, x(k)=5·3^{k-1}-4·2^{k-1}.

## Answer

**Partial fractions of X(z)/z.**

X(z)/z = (z + 2)/[z (z − 2)(z − 3)] = A/z + B/(z − 2) + C/(z − 3)

```text
A = (0 + 2)/[(0 − 2)(0 − 3)] = 2/6  = 1/3
B = (2 + 2)/[2 (2 − 3)]      = 4/(−2) = −2
C = (3 + 2)/[3 (3 − 2)]      = 5/3
```

X(z) = 1/3 − 2 z/(z − 2) + (5/3) z/(z − 3)

Inverse: the constant 1/3 gives (1/3)δ(k); z/(z − a) gives aᵏ.

**x(k) = (1/3) δ(k) − 2·2ᵏ + (5/3)·3ᵏ**

For k ≥ 1 this simplifies to the neat form

**Answer:** x(k) = 5·3^{k−1} − 4·2^{k−1} (k ≥ 1), and x(0) = 0.

Check: expanding X(z) in powers of z⁻¹ gives x(0..4) = 0, 1, 7, 29, 103, and the formula gives 5·3^{k−1} − 4·2^{k−1} = 1, 7, 29, 103 for k = 1..4 ✓.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. X(z) = (z+2)/((z-2)(z-3)) divided by z gives a 1/z factor, producing
   A. an A/z term, i.e. a constant in X(z)
   B. no extra term
   C. a double pole
   D. a z^2 term
   Answer: A. A/z becomes A. (Key point: Why divide)

2. A, B, C are
   A. 1/3, 2, 5/3
   B. 1/3, -2, 5/3
   C. 2, -2, 5
   D. -1/3, -2, 5/3
   Answer: B. Cover-up at z = 0, 2, 3. (Key point: Coefficients)

3. A constant in X(z) inverts to
   A. a constant sequence
   B. a step
   C. a delta(k) impulse at k = 0
   D. a^k
   Answer: C. Z[delta(k)] = 1. (Key point: Constant term)

4. For k >= 1, x(k) =
   A. 5*3^k - 4*2^k
   B. 2^k - 3^k
   C. 3^(k-1) + 2^(k-1)
   D. 5*3^(k-1) - 4*2^(k-1), and x(0) = 0
   Answer: D. Combine and drop impulse; x(1)=1. (Key point: Result)

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