Inverse Z-transform of function

Short answer

Use partial fractions on X(z)/z to get X(z)=z/(z-2)-z/(z-1). Apply inverse Z-transform of z/(z-a) being a^k to obtain x(k)=2^k - 1 for k=0,1,2,…

Find the inverse z-transform of z/((z − 1)(z − 2)).

Use partial fractions on X(z)/z.

X(z)/z = 1/((z − 1)(z − 2)) = 1/(z − 2) − 1/(z − 1)
X(z)   = z/(z − 2) − z/(z − 1)

Using Z⁻¹[z/(z − a)] = aᵏ:

Answer: x(k) = 2ᵏ − 1, k = 0, 1, 2, ...

Check by long division: X(z) = z/(z² − 3z + 2) = z⁻¹ + 3z⁻² + 7z⁻³ + ..., so x(0) = 0, x(1) = 1, x(2) = 3, x(3) = 7. These equal 2ᵏ − 1 ✓.

✓ Answer checked by Claude Opus 5.5 (high effort)

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