---
title: "z-transform of k²aᵏ"
subject: "Applied Mathematics"
chapter: 2
key: "semester_4/applied_mathematics:z-transform-k2-a-k"
source: https://www.bctengineer.com/q/semester_4/applied_mathematics/z-transform-k2-a-k
site: BCT Engineering
---

# z-transform of k²aᵏ

Applied Mathematics · Chapter 2: The Z-Transform

- Asked in: 2069 Poush
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/applied_mathematics/z-transform-k2-a-k
- Whole chapter: https://www.bctengineer.com/md/semester_4/applied_mathematics/chapter-2

## Question

Find the z-transform of k² aᵏ for k ≥ 0.

## Short answer

Apply Z[k x(k)] = -z d/dz X(z) twice, starting from Z[a^k]=z/(z-a). Result: Z[k^2 a^k] = a z (z+a)/(z-a)^3, |z|>|a|.

## Answer

**Method.** Use the property Z[k·x(k)] = −z d/dz X(z), applied twice, starting from Z[aᵏ] = z/(z − a).

**Step 1.** X₀(z) = Z[aᵏ] = z/(z − a)

**Step 2.** Z[k aᵏ] = −z d/dz [z/(z − a)]

d/dz [z/(z − a)] = −a/(z − a)²

so Z[k aᵏ] = az/(z − a)²

**Step 3.** Z[k² aᵏ] = −z d/dz [az/(z − a)²]

```text
d/dz [ a z (z − a)⁻² ] = a (z − a)⁻² − 2a z (z − a)⁻³
                       = a [ (z − a) − 2z ] / (z − a)³
                       = −a (z + a) / (z − a)³

Z[k² aᵏ] = −z · ( −a (z + a)/(z − a)³ )
```

**Answer:** Z[k² aᵏ] = a z (z + a) / (z − a)³, |z| > |a|.

Check: a = 1 gives z(z + 1)/(z − 1)³ = Z[k²] ✓. (Numerical check a = 0.5, z = 3: 0.336 both ways.)

## Check yourself

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

1. Which property is applied twice?
   A. Z[k x(k)] = -z dX/dz
   B. Z[k x(k)] = z dX/dz
   C. Z[a^k x(k)] = X(z/a)
   D. Z[x(k+1)] = zX(z)
   Answer: A. Multiplying by k corresponds to -z d/dz. (Key point: Property used)

2. Z[k a^k] =
   A. z/(z - a)^2
   B. a z/(z - a)^2
   C. a z/(z - a)
   D. a^2 z/(z - a)^2
   Answer: B. -z * (-a/(z-a)^2). (Key point: First step)

3. d/dz[a z/(z-a)^2] simplifies to
   A. a(z + a)/(z - a)^3
   B. -a(z - a)/(z - a)^3
   C. -a(z + a)/(z - a)^3
   D. -a z/(z - a)^3
   Answer: C. a[(z-a) - 2z]/(z-a)^3 = -a(z+a)/(z-a)^3. (Key point: Differentiation)

4. Z[k^2 a^k] =
   A. a z (z - a)/(z + a)^3
   B. a^2 z (z + a)/(z - a)^3
   C. z (z + a)/(z - a)^3
   D. a z (z + a)/(z - a)^3, |z| > |a|
   Answer: D. Multiplying by -z; a = 1 gives Z[k^2]. (Key point: Result)