Inverse z of z/((z-2)(z-1))

Short answer

Use partial fractions: X(z)=z/(z-2)-z/(z-1). Inverse gives x(k)=2^k-1 for k≥0.

Obtain inverse Z-transform of z/((z - 2)(z - 1)).

Partial fractions of X(z)/z.

X(z)/z = 1/[(z − 2)(z − 1)] = A/(z − 2) + B/(z − 1)

A = 1/(2 − 1) = 1
B = 1/(1 − 2) = −1

X(z) = z/(z − 2) − z/(z − 1)

Using Z⁻¹[z/(z − a)] = aᵏ:

Answer: x(k) = 2ᵏ − 1, k ≥ 0.

Check: x(0) = 0, x(1) = 1, x(2) = 3, x(3) = 7, x(4) = 15. The series expansion of X(z) = z⁻¹/[(1 − 2z⁻¹)(1 − z⁻¹)] gives the same values.

✓ Answer checked by Claude Opus 5.5 (high effort)

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